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A154921
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Triangle read by rows, the coefficients in ascending order of x^i of the polynomials p{0}(x) = 1 and p{n}(x) = sum_{k=0..n-1} binomial(n,k)*p{k}(0)*(1+x^(n-k)).
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10
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1, 1, 1, 3, 2, 1, 13, 9, 3, 1, 75, 52, 18, 4, 1, 541, 375, 130, 30, 5, 1, 4683, 3246, 1125, 260, 45, 6, 1, 47293, 32781, 11361, 2625, 455, 63, 7, 1, 545835, 378344, 131124, 30296, 5250, 728, 84, 8, 1
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OFFSET
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1,4
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COMMENTS
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Old name was: Matrix inverse of A154926.
A000670 appears in the first column. A052882 appears in the second column. A000027 and A045943 appear as diagonals. An alternative to calculating the matrix inverse of A154926 is to move the term in the lower right corner to a position in the same column and calculate the determinant instead, which yields the same answer.
Matrix inverse of (2*I - P), where P is Pascal's triangle and I the identity matrix. See A162312 for the matrix inverse of (2*P - I) and some general remarks about arrays of the form M(a) := (I - a*P)^-1 and their connection with weighted sums of powers of integers. The present array equals 1/2*M(1/2). - Peter Bala, Jul 01 2009
The values in this triangle can be seen as permanents of the Pascal triangle analogous to the method in the Redheffer matrix. The elements satisfy T(n,k)/T(n,k-1)*k=T(n-1,k)/T(n,k)*n which converges to log(2) as n-->infinity and k-->0. More generally to calculate log(x) multiply the negative values in A154926 with 1/(x-1) and calculate the matrix inverse. Then T(n,k)/T(n,k-1)*k and T(n-1,k)/T(n,k)*n in the resulting triangle converge to log(x). [From Mats Granvik, Aug 11 2009]
This method for calculating log(x) converges faster than the Taylor series when x is greater than 5 or so. See chapter on Taylor series in Spiegel for comparison. [From Mats Granvik, Aug 11 2009]
Exponential Riordan array [1/(2-exp(x)),x]. [Paul Barry, 6 April 2011]
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REFERENCES
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R. B. Nelsen, Problem E3062: Amer. Math. Monthly, Vol. 94, No. 4 (Apr., 1987), 376-377.
R. B. Nelsen and H. Schmidt, Jr., Chains in Power Sets, Mathematics Magazine, Vol. 64, No. 1 (Feb., 1991), 23-31.
Murray R. Spiegel, Mathematical handbook, Schaum's Outlines, p. 111.
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LINKS
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Table of n, a(n) for n=1..45.
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FORMULA
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Contribution from Peter Bala, Jul 01 2009: (Start)
TABLE ENTRIES
(1)... T(n,k) = binomial(n,k)*A000670(n-k).
GENERATING FUNCTION
(2)... exp(x*t)/(2-exp(t)) = 1 + (1+x)*t + (3+2*x+x^2)*t^2/2! + ....
PROPERTIES OF THE ROW POLYNOMIALS
The row generating polynomials R_n(x) form an Appell sequence. They
appear in the study of the poset of power sets [Nelsen and Schmidt].
The first few values are R_0(x) = 1, R_1(x) = 1+x, R_2(x) = 3+2*x+x^2
and R_3(x) = 13+9*x+3*x^2+x^3.
The row polynomials may be recursively computed by means of
(3)... R_n(x) = x^n + sum {k = 0..n-1} binomial(n,k)*R_k(x).
Explicit formulas include
(4)... R_n(x) = 1/2*sum {k = 0..inf}(1/2)^k*(x+k)^n,
(5)... R_n(x) = sum {j = 0..n} sum {k = 0..j} (-1)^(j-k)*binomial(j,k)
*(x+k)^n,
and
(6)... R_n(x) = sum {j = 0..n} sum{k = j..n} k!*Stirling2(n,k)
*binomial(x,k-j).
SUMS OF POWERS OF INTEGERS
The row polynomials satisfy the difference equation
(7)... 2*R_m(x) - R_m(x+1) = x^m,
which easily leads to the evaluation of the weighted sums of powers
of integers
(8)... sum {k = 1..n-1} (1/2)^k*k^m = 2*R_m(0) - (1/2)^(n-1)*R_m(n).
For example, m = 2 gives
(9)... sum {k = 1..n-1} (1/2)^k*k^2 = 6 - (1/2)^(n-1)*(n^2+2*n+3).
More generally we have
(10)... sum {k=0..n-1} (1/2)^k*(x+k)^m = 2*R_m(x)-(1/2)^(n-1)*R_m(x+n).
RELATIONS WITH OTHER SEQUENCES
Sequences in the database given by particular values of the row
polynomials are
(11)... A000670(n) = R_n(0)
(12)... A052841(n) = R_n(-1)
(13)... A000629(n) = R_n(1)
(14)... A007047(n) = R_n(2)
(15)... A080253(n) = 2^n*R_n(1/2).
This last result is the particular case (x = 0) of the result that
the polynomials 2^n*R_n(1/2+x/2) are the row generating polynomials
for A162313.
The above formulas should be compared with those for A162312.
(End)
... A151919 = R_n(1/3)*3^n*(-1)^n
... A052882 = [x^1]R_n(x)
... A045943 = [x^(n-1)]R_n+1(x)
... A099880 = [x^n]R_2n(x) - Peter Luschny, Jul 15 2012
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EXAMPLE
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Contribution from Peter Bala, Jul 01 2009: (Start)
Triangle begins
==============================================
n\k|.....0.....1.....2.....3.....4.....5.....6
==============================================
0..|.....1
1..|.....1.....1
2..|.....3.....2.....1
3..|....13.....9.....3.....1
4..|....75....52....18.....4.....1
5..|...541...375...130....30.....5.....1
6..|..4683..3246..1125...260....45.....6.....1
...
(End)
Contribution from Mats Granvik, Aug 11 2009: (Start)
Row 4 equals 75,52,18,4,1 because permanents of:
1,0,0,0,1..1,0,0,0,0..1,0,0,0,0..1,0,0,0,0..1,0,0,0,0
1,1,0,0,0..1,1,0,0,1..1,1,0,0,0..1,1,0,0,0..1,1,0,0,0
1,2,1,0,0..1,2,1,0,0..1,2,1,0,1..1,2,1,0,0..1,2,1,0,0
1,3,3,1,0..1,3,3,1,0..1,3,3,1,0..1,3,3,1,1..1,3,3,1,0
1,4,6,4,0..1,4,6,4,0..1,4,6,4,0..1,4,6,4,0..1,4,6,4,1
are:
75.........52.........18.........4..........1
(End)
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MAPLE
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A154921_row := proc(n) local i, p; p := proc(n, x) option remember; local k;
if n = 0 then 1 else add(p(k, 0)*binomial(n, k)*(1+x^(n-k)), k=0..n-1) fi end:
seq(coeff(p(n, x), x, i), i=0..n) end: for n from 0 to 5 do A154921_row(n) od;
# Peter Luschny, Jul 15 2012
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PROG
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(Sage)
@CachedFunction
def Poly(n, x) : return 1 if n == 0 else add(Poly(k, 0)*binomial(n, k)*(x^(n-k)+1) for k in range(n)
R = PolynomialRing(ZZ, 'x')
for n in (0..6) : print R(Poly(n, x)).coeffs() # Peter Luschny, Jul 15 2012
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CROSSREFS
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A000629 (row sums), A000670, A007047, A052822 (column 1), A052841 (alt. row sums), A080253, A162312, A162313.
Sequence in context: A134090 A132845 A129652 * A127126 A161133 A112911
Adjacent sequences: A154918 A154919 A154920 * A154922 A154923 A154924
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KEYWORD
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nonn,tabl
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AUTHOR
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Mats Granvik, Jan 17 2009
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EXTENSIONS
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New name gives a reference-free definition. - Peter Luschny, Jul 15 2012
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STATUS
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approved
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