Search: a077028
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A077028
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The rascal triangle, read by rows: T(n,k) (n >= 0, 0 <= k <= n) = k(n-k)+1.
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+30
18
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1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 4, 5, 4, 1, 1, 5, 7, 7, 5, 1, 1, 6, 9, 10, 9, 6, 1, 1, 7, 11, 13, 13, 11, 7, 1, 1, 8, 13, 16, 17, 16, 13, 8, 1, 1, 9, 15, 19, 21, 21, 19, 15, 9, 1, 1, 10, 17, 22, 25, 26, 25, 22, 17, 10, 1, 1, 11, 19, 25, 29, 31, 31, 29, 25, 19, 11, 1, 1, 12, 21, 28, 33, 36
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OFFSET
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0,5
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COMMENTS
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Pascal's triangle is formed using the rule South = West + East, whereas the rascal triangle uses the rule South = (West*East+1)/North. [Anggoro et al.]
The n-th diagonal is congruent to 1 mod n-1.
Row sums are the cake numbers, A000125. Alternating sum of row n is 0 if n even and (3-n)/2 if n odd. Rows are symmetric, beginning and ending with 1. The number of occurrences of k in this triangle is the number of divisors of k-1, given by A000005.
The triangle can be generated by numbers of the form k*(n-k) + 1 for k = 0 to n. Conjecture: except for n = 0,1 and 6 every row contains a prime. - Amarnath Murthy, Jul 15 2005
Comments from Moshe Shmuel Newman, Apr 06 2008: (Start) Consider the semigroup of words in x,y,q subject to the relationships: yx = xyq, qx = xq, qy = yq
Now take words of length n in x and y, with exactly k y's. If there had been no relationships, the number of different words of this type would be n choose k, sequence A007318. Thanks to the relationships, the number of words of this type is the k-th entry in the n-th row of this sequence (read as a triangle, with the first row indexed by zero and likewise the first entry in each row.)
For example: with three letters and one y, we have three possibilities: xxy, xyx = xxyq, yxx = xxyqq. No two of them are equal, so this entry is still 3, as in Pascal's triangle.
With four letters, two y's, we have the first reduction: xyyx = yxxy = xxyyqq and this is the only reduction for 4 letters. So the middle entry of the fourth row is 5 instead of 6, as in the Pascal triangle. (End)
Main diagonals of this triangle sum to polygonal numbers. See A057145. - Raphie Frank, Oct 30 2012
T(n,k) gives the number of distinct sums of k elements in {1,2,... n}, e.g., T(5,4) = the number of distinct sums of 4 elements in {1,2,3,4,5}, which is (5+4+3+2) - (4+3+2+1) + 1 = 5. - Derek Orr, Nov 26 2014
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LINKS
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Table of n, a(n) for n=0..83.
A. Anggoro, E. Liu and A. Tulloch, The Rascal Triangle, College Math. J., Vol. 41, No. 5, Nov. 2010, pp. 393-395.
Brian Hopkins, Editorial: Anonymity and Youth, The College Mathematics Journal, 45 (Number 2, 2014), 82. - From N. J. A. Sloane, Apr 05 2014
L. McHugh, CMJ Article Shows Collaboration Is Not Limited by Geography ... or Age, MAA Focus (Magazine), Vol. 31, No. 1, 2011, p. 13.
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FORMULA
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As a square array read by antidiagonals, a(n, k) = 1+n*k. a(n, k)=a(n-1, k)+k. Row n has g.f. (1+(n-1)x)/(1-x)^2, n>=0. - Paul Barry, Feb 22 2003
Still thinking of square arrays. Let f:N->Z and g:N->Z be given and I an integer, then define a(n, k) = I + f(n)*g(k). Then a(n, k)*a(n-1, k-1)=a(n-1, k)*a(n, k-1) +I*(f(n)-f(n-1))*(g(k)-g(k-1)) for suitable n and k. S= (E*W +1)/N. arises with I = 1, and f = g = id. [Terry Lindgren, Apr 10 2011]
T(n,k) = A128139(n-1,k-1). - Gary W. Adamson, Jul 02 2012
O.g.f. (1 - x*(1 + t) + 2*t*x^2)/((1 - x)^2*(1 - t*x)^2) = 1 + (1 + t)*x + (1 + 2*t + t^2)*x^2 + .... Cf. A105851. - Peter Bala, Jul 26 2015
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EXAMPLE
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Third diagonal (1,3,5,7,...) consists of the positive integers congruent to 1 mod 2.
Triangle begins:
1
1 1
1 2 1
1 3 3 1
1 4 5 4 1
1 5 7 7 5 1
1 6 9 10 9 6 1
...
As a square array read by antidiagonals, the first rows are:
1 1 1. 1. 1. 1 ...
1 2 3. 4. 5. 6 ...
1 3 5. 7. 9 11 ...
1 4 7 10 13 16 ...
1 5 9 13 17 21 ...
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MATHEMATICA
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t[n_, k_] := k (n - k) + 1; t[0, 0] = 1; Table[ t[n, k], {n, 0, 12}, {k, 0, n}] // Flatten (* Robert G. Wilson v, Jul 06 2012 *)
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PROG
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(PARI) {T(n, k) = if( k<0 | k>n, 0, k * (n - k) + 1)} /* Michael Somos, Mar 20 2011 */
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CROSSREFS
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Cf. A077029, A003991, A105851.
The maximum value for each anti-diagonal is given by sequence A033638.
Equals A004247(n) + 1.
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KEYWORD
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nonn,tabl
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AUTHOR
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Clark Kimberling, Oct 19 2002
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EXTENSIONS
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Better definition based on Murthy's comment of Jul 15 2005 and the Anggoro et al. paper. - N. J. A. Sloane, Mar 05 2011
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STATUS
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approved
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1, 1, 1, 1, 3, 1, 1, 5, 5, 1, 1, 7, 10, 7, 1, 1, 9, 16, 16, 9, 1, 1, 11, 23, 29, 23, 11, 1, 1, 13, 31, 47, 47, 31, 13, 1, 1, 15, 40, 71, 86, 71, 40, 15, 1, 1, 17, 50, 102, 146, 146, 102, 50, 17, 1, 1, 19, 61, 141, 234, 277, 234, 141, 61, 19, 1
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OFFSET
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1,5
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COMMENTS
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Row sums = A116725: (1, 2, 5, 12, 26, 52,...).
Row sums are {1, 2, 5, 12, 26, 52, 99, 184, 340, 632, 1189}. [From Roger L. Bagula, Nov 02 2008]
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LINKS
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Table of n, a(n) for n=1..66.
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FORMULA
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p(x,n)=If[n == 0, 1, (x + 1)^n + Sum[(n - m)*(m)*x^m*(1 + x^(n - 2*m)), {m, 1, n - 1}]/2]; t(n,m)=Coefficients(p(x,n)) [From Roger L. Bagula, Nov 02 2008]
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EXAMPLE
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First few rows of the triangle are:
1;
1, 1;
1, 3, 1;
1, 5, 5, 1;
1, 7, 10, 7, 1;
1, 9, 16, 16, 9, 1;
1, 11, 23, 29, 23, 11, 1;
1, 13, 31, 47, 47, 31, 13, 1;
1, 15, 40, 71, 86, 71, 40, 15, 1;
...
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MATHEMATICA
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Clear[p, x, n]; p[x_, n_] = If[ n == 0, 1, (x + 1)^n +Sum[(n - m)*(m)*x^m*(1 + x^(n - 2*m)), {m, 1, n - 1}]/2]; Table[CoefficientList[FullSimplify[ExpandAll[p[x, n]]], x], {n, 0, 10}]; Flatten[%] [From Roger L. Bagula, Nov 02 2008]
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CROSSREFS
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Cf. A077028, A116725.
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KEYWORD
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nonn,tabl
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AUTHOR
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Gary W. Adamson, Oct 23 2007
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EXTENSIONS
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Extended by Roger L. Bagula, Nov 02 2008
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STATUS
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approved
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A134394
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Triangle T(n,k) = Sum_{j=k..n} A077028(j,k), read by rows.
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+20
2
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1, 2, 1, 3, 3, 1, 4, 6, 4, 1, 5, 10, 9, 5, 1, 6, 15, 16, 12, 6, 1, 7, 21, 25, 22, 15, 7, 1, 8, 28, 36, 35, 28, 18, 8, 1, 9, 36, 49, 51, 45, 34, 21, 9, 1, 10, 45, 64, 70, 66, 55, 40, 24, 10, 1, 11, 55, 81, 92, 91, 81, 65, 46, 27, 11, 1, 12, 66, 100, 117, 120, 112, 96, 75, 52, 30, 12, 1, 13, 78, 121, 145, 153
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OFFSET
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1,2
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COMMENTS
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Row sums = A055795: (1, 3, 7, 15, 30, 56, 98, ...).
Antidiagonal reading of A139600 without its left column. -_ R. J. Mathar_, Apr 17 2011
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LINKS
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Table of n, a(n) for n=1..83.
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FORMULA
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A000012 * A077028 as infinite lower triangular matrices.
T(n,k) = (k-n-1)*( k*(k-n)-2)/2. - R. J. Mathar, Apr 17 2011
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EXAMPLE
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First few rows of the triangle are:
1;
2, 1;
3, 3, 1;
4, 6, 4, 1;
5, 10, 9, 5, 1;
6, 15, 16, 12, 6, 1;
7, 21, 25, 22, 15, 7, 1;
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MAPLE
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A077028 := proc(n, k) if n < 0 or k<0 or k > n then 0; else k*(n-k)+1 ; end if; end proc:
A134394 := proc(n, k) add ( A077028(j, k), j=k..n) ; end proc:
seq(seq(A134394(n, k), k=0..n), n=0..15) ; # R. J. Mathar, Apr 17 2011
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CROSSREFS
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Cf. A077028, A055795.
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KEYWORD
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nonn,tabl
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AUTHOR
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Gary W. Adamson, Oct 23 2007
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STATUS
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approved
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1, 2, 1, 4, 4, 1, 8, 12, 6, 1, 16, 32, 23, 8, 1, 32, 80, 72, 37, 10, 1, 64, 192, 201, 132, 54, 12, 1, 128, 448, 522, 405, 216, 74, 14, 1, 256, 1024, 1291, 1128, 723, 328, 97, 16, 1, 512, 2304, 3084, 2941, 2154, 1191, 472, 123, 18, 1, 1024, 5120, 7181, 7316, 5920
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OFFSET
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0,2
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COMMENTS
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Row sums = A134396: (1, 3, 9, 27, 80, 232, 656,...).
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LINKS
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Table of n, a(n) for n=0..59.
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FORMULA
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A007318 * A077028 as infinite lower triangular matrices.
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EXAMPLE
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First few rows of the triangle are:
1;
2, 1;
4, 4, 1;
8, 12, 6, 1;
16, 32, 23, 8, 1;
32, 80, 72, 37, 10, 1;
...
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MAPLE
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A007318 := proc(n, k) binomial(n, k) ; end: A077028 := proc(i, j) if j <= i then (i-j)*(j-1)+1 ; else 0 ; fi ; end: A134395 := proc(n, m) add(A007318(n, k)*A077028(k+1, m+1), k=0..n) ; end: for n from 0 to 15 do for m from 0 to n do printf("%d, ", A134395(n, m)) ; od: od: # R. J. Mathar, Jun 08 2008
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CROSSREFS
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Cf. A077028, A134396.
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KEYWORD
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nonn,tabl
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AUTHOR
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Gary W. Adamson, Oct 23 2007
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EXTENSIONS
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Corrected and extended by R. J. Mathar, Jun 08 2008
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STATUS
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approved
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1, 1, 2, 1, 5, 4, 1, 10, 17, 8, 1, 19, 51, 49, 16, 1, 36, 134, 196, 129, 32, 1, 69, 330, 650, 645, 321, 64, 1, 134, 783, 1940, 2575, 1926, 769, 128
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OFFSET
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1,3
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COMMENTS
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Let T(r,c) be the array A077028. Fill 2^k numbers in Gaussian templates conforming to the row lengths determined by T(r,c). A110552 results from summing the numbers on each row.
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REFERENCES
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P. Barry, A Note on a Family of Generalized Pascal Matrices Defined by Riordan Arrays, Journal of Integer Sequences, 16 (2013), #13.5.4.
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LINKS
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Table of n, a(n) for n=1..36.
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FORMULA
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Table entries appear to be given by T(n,k) = binomial(n-2,k-1) + 2^(n-1)*binomial(n-2,k-2), n,k >= 1, leading to the e.g.f. (exp((1+x)*u) - 1)*(x*exp((1+x)*u) + x + 2)/(2*(1+x)^2) = u + (1+2*x)*u^2/2! + (1+5*x+4*x^2)*u^3/3! + .... Cf. A111049. - Peter Bala, Jul 27 2012
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EXAMPLE
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The filled templates begin
1
.1
.2
..1
..2.3
..4
....1
....2.3.5
....4.6.7
....8
therefore the sequence begins
1
1 2
1 5 4
1 10 17 8
...
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CROSSREFS
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Cf. A077028, A007051, A007582. A111049.
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KEYWORD
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nonn,tabl
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AUTHOR
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Alford Arnold, Jul 26 2005
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STATUS
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approved
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A000125
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Cake numbers: maximal number of pieces resulting from n planar cuts through a cube (or cake): C(n+1,3)+n+1.
(Formerly M1100 N0419)
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+10
63
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1, 2, 4, 8, 15, 26, 42, 64, 93, 130, 176, 232, 299, 378, 470, 576, 697, 834, 988, 1160, 1351, 1562, 1794, 2048, 2325, 2626, 2952, 3304, 3683, 4090, 4526, 4992, 5489, 6018, 6580, 7176, 7807, 8474, 9178, 9920, 10701, 11522, 12384, 13288, 14235, 15226
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OFFSET
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0,2
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COMMENTS
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Note that a(n) = a(n-1) + A000124(n-1). This has the following geometrical interpretation: Define a number of planes in space to be in general arrangement when
(1) no two planes are parallel,
(2) there are no two parallel intersection lines,
(3) there is no point common to four or more planes.
Suppose there are already n-1 planes in general arrangement, thus defining the maximal number of regions in space obtainable by n-1 planes and now one more plane is added in general arrangement. Then it will cut each of the n-1 planes and acquire intersection lines which are in general arrangement. (See the comments on A000124 for general arrangement with lines.) These lines on the new plane define the maximal number of regions in 2-space definable by n-1 straight lines, hence this is A000124(n-1). Each of this regions acts as a dividing wall, thereby creating as many new regions in addition to the a(n-1) regions already there, hence a(n)=a(n-1)+A000124(n-1). - Peter C. Heinig (algorithms(AT)gmx.de), Oct 19 2006
More generally, we have: A000027(n) = binomial(n,0) + binomial(n,1) (the natural numbers), A000124(n) = binomial(n,0) + binomial(n,1) + binomial(n,2) (the Lazy Caterer's sequence), a(n) = binomial(n,0) + binomial(n,1) + binomial(n,2) + binomial(n,3) (Cake Numbers). - Peter C. Heinig (algorithms(AT)gmx.de), Oct 19 2006
If Y is a 2-subset of an n-set X then, for n>=3, a(n-3) is the number of 3-subsets of X which do not have exactly one element in common with Y. - Milan Janjic, Dec 28 2007
a(n) is the number of compositions (ordered partitions) of n+1 into four or fewer parts or equivalently the sum of the first four terms in the n-th row of Pascal's triangle. - Geoffrey Critzer, Jan 23 2009
{a(k): 0 <= k < 4} = divisors of 8. - Reinhard Zumkeller, Jun 17 2009
a(n) is also the maximum number of different values obtained by summing n consecutive positive integers with all possible 2^n sign combinations. This maximum is first reached when summing the interval [n, 2n-1]. - Olivier Gérard, Mar 22 2010
a(n) contains only 5 perfect squares > 1: 4, 64, 576, 676000, and 75203584. The incidences of > 0 are given by A047694. - Frank M Jackson, Mar 15 2013
Given n tiles with two values - an A value and a B value - a player may pick either the A value or the B value. The particular tiles are [n, 0], [n-1, 1], ..., [2, n-2] and [1, n-1]. The sequence is the number of different final A:B counts. For example, with n=4, we can have final total [5, 3] = [4, _] + [_, 1] + [_, 2] + [1, _] = [_, 0] + [3, _] + [2, _] + [_, 3], so a(4) = 2^4 - 1 = 15. The largest and smallest final A+B counts are given by A077043 and A002620 respectively. - Jon Perry, Oct 24 2014
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REFERENCES
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V. I. Arnold (ed.), Arnold's Problems, Springer, 2004, comments on Problem 1990-11 (p. 75), pp. 503-510. Numbers N_3.
R. B. Banks, Slicing Pizzas, Racing Turtles and Further Adventures in Applied Mathematics, Princeton Univ. Press, 1999. See p. 27.
L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 72, Problem 2.
H. E. Dudeney, Amusements in Mathematics, Nelson, London, 1917, page 177.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
T. H. Stickels, Mindstretching Puzzles. Sterling, NY, 1994 p. 85.
W. A. Whitworth, DCC Exercises in Choice and Chance, Stechert, NY, 1945, p. 30.
A. M. Yaglom and I. M. Yaglom: Challenging Mathematical Problems with Elementary Solutions. Vol. I. Combinatorial Analysis and Probability Theory. New York: Dover Publications, Inc., 1987, p. 13, #45 (First published: San Francisco: Holden-Day, Inc., 1964)
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LINKS
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T. D. Noe, Table of n, a(n) for n=0..1000
A. M. Baxter, L. K. Pudwell, Ascent sequences avoiding pairs of patterns, 2014.
D. A. Lind, On a class of nonlinear binomial sums, Fib. Quart., 3 (1965), 292-298.
Svante Linusson, The number of M-sequences and f-vectors, Combinatorica, vol 19 no 2 (1999) 255-266.
Alexsandar Petojevic, The Function vM_m(s; a; z) and Some Well-Known Sequences, Journal of Integer Sequences, Vol. 5 (2002), Article 02.1.7
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992.
Simon Plouffe, 1031 Generating Functions and Conjectures, Université du Québec à Montréal, 1992.
D. J. Price, Some unusual series occurring in n-dimensional geometry, Math. Gaz., 30 (1946), 149-150.
L. Pudwell, A. Baxter, Ascent sequences avoiding pairs of patterns, 2014.
Luis Manuel Rivera, Integer sequences and k-commuting permutations, arXiv preprint arXiv:1406.3081 [math.CO], 2014
H. P. Robinson, Letter to N. J. A. Sloane, Aug 16 1971, with attachments
Eric Weisstein's World of Mathematics, Cake Number
Eric Weisstein's World of Mathematics, Cube Division by Planes
Eric Weisstein's World of Mathematics, Cylinder Cutting
Eric Weisstein's World of Mathematics, Space Division by Planes
R. Zumkeller, Enumerations of Divisors [From Reinhard Zumkeller, Jun 17 2009]
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).
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FORMULA
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a(n) = (n+1)*(n^2-n+6)/6 = (n^3 + 5*n + 6) / 6.
G.f.: (1-2*x+2x^2)/(1-x)^4; - [Simon Plouffe in his 1992 dissertation.]
E.g.f.: (1+x+x^2/2+x^3/6)*exp(x).
a(n) = binomial(n,3)+binomial(n,2)+binomial(n,1)+binomial(n,0). [Peter C. Heinig (algorithms(AT)gmx.de), Oct 19 2006]
Paraphrasing the previous comment: the sequence is the binomial transform of [1,1,1,1,0,0,0,...]. - Gary W. Adamson, Oct 23 2007
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EXAMPLE
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a(4)=15 because there are 15 compositions of 5 into four or fewer parts. a(6)=42 because the sum of the first four terms in the 6th row of Pascal's triangle is 1+6+15+20=42. - Geoffrey Critzer, Jan 23 2009
For n=5, (1, 3, 5, 7, 9, 11, 13, 17, 19, 21, 23, 25, 35) and their opposite are the 26 different sums obtained by summing 5,6,7,8,9 with any sign combination. - Olivier Gérard, Mar 22 2010
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MAPLE
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A000125 := n->(n+1)*(n^2-n+6)/6;
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MATHEMATICA
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Table[(n^3+5n+6)/6, {n, 0, 50}] (* or *) LinearRecurrence[{4, -6, 4, -1}, {1, 2, 4, 8}, 50] (* Harvey P. Dale, Jan 19 2013 *)
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PROG
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(PARI) a(n)=(n^2+5)*n/6+1 \\ Charles R Greathouse IV, Jun 15 2011
(MAGMA) [(n^3+5*n+6)/6: n in [0..50]]; // Vincenzo Librandi, Nov 08 2014
(PARI) Vec((1-2*x+2*x^2)/((1-x)^4) + O(x^100)) \\ Altug Alkan, Oct 16 2015
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CROSSREFS
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Cf. A000124, A003600.
Bisections give A100503, A100504.
Row sums of A077028.
A005408, A000124, A016813, A086514, A058331, A002522, A161701 - A161705, A000127, A161706 - A161708, A080856, A161710 - A161713, A161715, A006261. - Reinhard Zumkeller, Jun 17 2009
Cf. A063865. - Olivier Gérard, Mar 22 2010
Cf. A051601. - Bruno Berselli, Aug 02 2013
Cf. A077043, A002620.
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KEYWORD
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nonn,easy,nice
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AUTHOR
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N. J. A. Sloane
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EXTENSIONS
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More terms from James A. Sellers, Feb 22 2000
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STATUS
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approved
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A083906
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Table T(n,k) read along rows: the coefficient [q^k] of the sum_{m=0..n} [n,m]_q over q-Binomial coefficients.
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+10
8
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1, 2, 3, 1, 4, 2, 2, 5, 3, 4, 3, 1, 6, 4, 6, 6, 6, 2, 2, 7, 5, 8, 9, 11, 9, 7, 4, 3, 1, 8, 6, 10, 12, 16, 16, 18, 12, 12, 8, 6, 2, 2, 9, 7, 12, 15, 21, 23, 29, 27, 26, 23, 21, 15, 13, 7, 4, 3, 1, 10, 8, 14, 18, 26, 30, 40, 42, 48, 44, 46, 40
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OFFSET
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0,2
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COMMENTS
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There are A033638(n) values in the n-th row, compliant with the order of the polynomial.
In the example for n=6 detailed below, the orders of [6,k]_q are 1,6,9,10,9,6,1 for k=0..6,
the maximum order 10 defining the row length
Note also that 1 6 9 10 9 6 1 and related distributions are antidiagonals of A077028.
A083480 is a variation illustrating a relationship with numeric partitions, A000041.
The rows are formed by the nonzero entries of the columns of A049597.
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REFERENCES
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Andrews(1976) Theory of Partitions (page 242)
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LINKS
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Table of n, a(n) for n=0..70.
Eric Weisstein, q-Binomial Coefficient, Mathworld.
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FORMULA
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Row sums: sum_k T(n,k)= 2^n.
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EXAMPLE
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When viewed as an array with A033638(r) entries per row, the table begins:
. 1 ....... : 1
. 2 ....... : 2
. 3 1 ....... : 3+q = (1)+(1+q)+(1)
. 4 2 2 ....... : 4+2q+2q^2 = 1+(1+q+q^2)+(1+q+q^2)+1
. 5 3 4 3 1 ....... : 5+3q+4q^2+3q^3+q^4
. 6 4 6 6 6 2 2
. 7 5 8 9 11 9 7 4 3 1 ....... : 7+5q+8q^2+9q^3+11q^4+9q^5+...
. 8 6 10 12 16 16 18 12 12 8 6 2 2
. 9 7 12 15 21 23 29 27 26 23 21 15 13 7 4 3 1
...
The second but last row is from the sum over 7 q-polynomials coefficients ....... :
. 1 ....... : 1 = [6,0]_q
. 1 1 1 1 1 1 ....... : 1+q+q^2+q^3+q^4+q^5 = [6,1]_q
. 1 1 2 2 3 2 2 1 1 ....... : 1+q+2q^2+2q^3+3q^4+2q^5+2q^6+q^7+q^8 = [6,2]_q
. 1 1 2 3 3 3 3 2 1 1 ....... : 1+q+2q^2+3q^3+3q^4+3q^5+3q^6+2q^7+q^8+q^9 = [6,3]_q
. 1 1 2 2 3 2 2 1 1 ....... : 1+q+2q^2+2q^3+3q^4+2q^5+2q^6+q^7+q^8 = [6,4]_q
. 1 1 1 1 1 1 ....... : 1+q+q^2+q^3+q^4+q^5 = [6,5]_q
. 1 ....... : 1 = [6,6]_q
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MAPLE
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QBinomial := proc(n, m, q) local i ; factor( mul((1-q^(n-i))/(1-q^(i+1)), i=0..m-1) ) ; expand(%) ; end:
A083906 := proc(n, k) add( QBinomial(n, m, q), m=0..n ) ; coeftayl(%, q=0, k) ; end:
for n from 0 to 10 do for k from 0 to A033638(n)-1 do printf("%d, ", A083906(n, k)) ; od: od: # R. J. Mathar, May 28 2009
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CROSSREFS
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Cf. A033638, A077028, A083479, A083480.
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KEYWORD
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nonn,tabf
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AUTHOR
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Alford Arnold, Jun 19 2003
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EXTENSIONS
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Edited by R. J. Mathar, May 28 2009
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STATUS
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approved
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A114219
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Number triangle (k-(k-1)*0^(n-k))*[k<=n].
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+10
5
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1, 0, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 2, 3, 1, 0, 1, 2, 3, 4, 1, 0, 1, 2, 3, 4, 5, 1, 0, 1, 2, 3, 4, 5, 6, 1, 0, 1, 2, 3, 4, 5, 6, 7, 1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 1
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OFFSET
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0,9
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COMMENTS
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Row sums are n(n-1)/2+1 (essentially A000124). Diagonal sums are A114220. First difference triangle of A077028, when this is viewed as a number triangle.
The matrix inverse is
1;
0,1;
0,-1,1;
0,1,-2,1;
0,-2,4,-3,1;
0,6,-12,9,-4,1;
0,-24,48,-36,16,-5,1;
0,120,-240,180,-80,25,-6,1;
0,-720,1440,-1080,480,-150,36,-7,1;
.. apparently related to A208058. - R. J. Mathar, Mar 22 2013
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LINKS
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Table of n, a(n) for n=0..65.
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EXAMPLE
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Triangle begins
1;
0, 1;
0, 1, 1;
0, 1, 2, 1;
0, 1, 2, 3, 1;
0, 1, 2, 3, 4, 1;
0, 1, 2, 3, 4, 5, 1;
0, 1, 2, 3, 4, 5, 6, 1;
0, 1, 2, 3, 4, 5, 6, 7, 1;
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MAPLE
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A114219 := proc(n, k)
if k < 0 or k > n then
0;
elif n = k then
1;
else
k ;
end if;
end proc: # R. J. Mathar, Mar 22 2013
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KEYWORD
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easy,nonn,tabl
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AUTHOR
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Paul Barry, Nov 18 2005
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STATUS
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approved
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1, 2, 6, 20, 64, 192, 544, 1472, 3840, 9728, 24064, 58368, 139264, 327680, 761856, 1753088, 3997696, 9043968, 20316160, 45350912, 100663296, 222298112, 488636416, 1069547520, 2332033024, 5066719232, 10972299264, 23689428992
(list;
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OFFSET
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0,2
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LINKS
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Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (6,-12,8).
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FORMULA
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G.f.: (1-4*x+6*x^2)/(1-2*x)^3. [Colin Barker, Apr 01 2012]
a(n) = 6*a(n-1) - 12*a(n-2) + 8*a(n-3). _ Vincenzo Librandi_, Apr 28 2012
a(n) = Sum_{k=0..n} binomial(n,k) * A077028(n,k), where A077028(n,k) = (n-k)*k + 1. - Paul D. Hanna, Oct 11 2015
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MATHEMATICA
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CoefficientList[Series[(1-4*x+6*x^2)/(1-2*x)^3, {x, 0, 30}], x] (* Vincenzo Librandi, Apr 28 2012 *)
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PROG
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(MAGMA) I:=[1, 2, 6]; [n le 3 select I[n] else 6*Self(n-1)-12*Self(n-2)+8*Self(n-3): n in [1..30]]; // Vincenzo Librandi, Apr 28 2012
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CROSSREFS
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Cf. A053545.
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KEYWORD
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nonn,easy
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AUTHOR
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N. J. A. Sloane, Mar 24 2000
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STATUS
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approved
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A105851
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Binomial transform triangle, read by rows.
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+10
2
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1, 2, 1, 4, 3, 1, 8, 8, 4, 1, 16, 20, 12, 5, 1, 32, 48, 32, 16, 6, 1, 64, 112, 80, 44, 20, 7, 1, 128, 256, 192, 112, 56, 24, 8, 1, 256, 576, 448, 272, 144, 68, 28, 9, 1, 512, 1280, 1024, 640, 352, 176, 80, 32, 10, 1, 1024, 2816, 2304, 1472, 832, 432, 208, 92, 36, 11, 1
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OFFSET
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0,2
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COMMENTS
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Let P = Pascal's triangle as an infinite lower triangular matrix and A = the infinite array of arithmetic sequences as shown in A077028:
1 1 1 1 1...
1 2 3 4 5...
1 3 5 7 9...
1 4 7 10 13...
1 5 9 13 17...
Perform the operation P * A, getting a new array with each column being the binomial transform of an arithmetic sequence. Take antidiagonals of the new array, then by rows = the triangle of A105851.
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LINKS
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Table of n, a(n) for n=0..65.
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FORMULA
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n-th column of the triangle is the binomial transform of the arithmetic sequence (n*k + 1), (k = 0, 1, 2...).
From Peter Bala, Jul 26 2015: (Start)
T(n,k) = (2 + k*(n - k))*2^(n-k-1) for 0 <= k <= n.
O.g.f. (1 - x*(2 + t) + 3*t*x^2)/((1 - 2*x)^2*(1 - t*x)^2) = 1 + (2 + t)*x + (4 + 3*t + t^2)*x^2 + ....
k-th column g.f. (1 + (k - 2)*x)/(1 - 2*x)^2. Cf. A077028. (End)
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EXAMPLE
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Column 3: 1, 5, 16, 44, 112...(A053220) is the binomial transform of 3k+1 (A016777: 1, 4, 7,...).
Triangle begins:
1;
2, 1;
4, 3, 1;
8, 8, 4, 1;
16, 20, 12, 5, 1;
32, 48, 32, 16, 6, 1;
64, 112, 80, 44, 20, 7, 1;
128, 256, 192, 112, 56, 24, 8, 1;
256, 576, 448, 272, 144, 68, 28, 9, 1;
512, 1280, 1024, 640, 352, 176, 80, 32, 10, 1;
1024, 2816, 2304, 1472, 832, 432, 208, 92, 36, 11, 1 ;...
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MAPLE
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seq(seq((2 + k*(n - k))*2^(n-k-1), k=0..n), n=0..10); - Peter Bala, Jul 26 2015
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MATHEMATICA
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t[n_, k_]:=(2 + k (n - k)) 2^(n - k - 1); Table[t[n - 1, k - 1], {n, 10}, {k, n}]//Flatten (* Vincenzo Librandi, Jul 26 2015 *)
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PROG
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(MAGMA) /* As triangle */ [[(2+k*(n-k))*2^(n-k-1): k in [0..n]]: n in [0.. 15]]; // Vincenzo Librandi, Jul 26 2016
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CROSSREFS
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Cf. A077028, A001792, A001787, A053220, A016777, A014480.
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KEYWORD
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nonn,tabl
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AUTHOR
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Gary W. Adamson, Apr 23 2005
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EXTENSIONS
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More terms from Philippe Deléham, Mar 31 2007
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STATUS
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approved
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