Search: seq:1,1,1,1,2,1,1,2,3,1
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A027751
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Irregular triangle read by rows in which row n lists the proper divisors of n (those divisors of n which are < n), with the first row {1} by convention.
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+30
59
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1, 1, 1, 1, 2, 1, 1, 2, 3, 1, 1, 2, 4, 1, 3, 1, 2, 5, 1, 1, 2, 3, 4, 6, 1, 1, 2, 7, 1, 3, 5, 1, 2, 4, 8, 1, 1, 2, 3, 6, 9, 1, 1, 2, 4, 5, 10, 1, 3, 7, 1, 2, 11, 1, 1, 2, 3, 4, 6, 8, 12, 1, 5, 1, 2, 13, 1, 3, 9, 1, 2, 4, 7, 14, 1, 1, 2, 3, 5, 6, 10, 15, 1, 1, 2, 4, 8, 16, 1, 3, 11, 1, 2, 17, 1, 5, 7, 1, 2, 3, 4, 6, 9, 12, 18
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OFFSET
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1,5
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COMMENTS
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Or, take the list 1,2,3,4,... of natural numbers (A000027) and replace each number by its proper divisors.
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LINKS
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EXAMPLE
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The irregular triangle T(n,k) begins:
n\k 1 2 3 4 5 ...
1: 1 (by convention)
2: 1
3: 1
4: 1 2
5: 1
6: 1 2 3
7: 1
8: 1 2 4
9: 1 3
10: 1 2 5
11: 1
12: 1 2 3 4 6
13: 1
14: 1 2 7
15: 1 3 5
16: 1 2 4 8
17: 1
18: 1 2 3 6 9
19: 1
20: 1 2 4 5 10
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MAPLE
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with(numtheory):
T:= n-> sort([(divisors(n) minus {n})[]])[]: T(1):=1:
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MATHEMATICA
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Table[ Divisors[n] // Most, {n, 1, 36}] // Flatten // Prepend[#, 1] & (* Jean-François Alcover, Jun 10 2013 *)
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PROG
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(Haskell)
a027751 n k = a027751_tabf !! (n-1) !! (k-1)
a027751_row n = a027751_tabf !! (n-1)
a027751_tabf = [1] : map init (tail a027750_tabf)
(Python)
from sympy import divisors
def a(n): return [1] if n==1 else divisors(n)[:-1]
(PARI) row(n) = if (n==1, [1], my(d = divisors(n)); vector(#d-1, k, d[k])); \\ Michel Marcus, Apr 30 2017
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CROSSREFS
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KEYWORD
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nonn,easy,tabf
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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A153452
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a(1) = 1; if n > 1, then a(n) = Sum_{prime q |n} a(n*q' /q), where q' = prevprime(q) for q>2 and 2' = 1.
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+30
58
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1, 1, 1, 1, 1, 2, 1, 1, 2, 3, 1, 3, 1, 4, 5, 1, 1, 5, 1, 6, 9, 5, 1, 4, 5, 6, 5, 10, 1, 16, 1, 1, 14, 7, 14, 9, 1, 8, 20, 10, 1, 35, 1, 15, 21, 9, 1, 5, 14, 21, 27, 21, 1, 14, 28, 20, 35, 10, 1, 35, 1, 11, 56, 1, 48, 64, 1, 28, 44, 70, 1, 14, 1, 12, 42, 36, 42
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OFFSET
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1,6
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COMMENTS
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Original name was: a(1)=1, for n>1, if 2*m = n or (m/p)*nextprime(p) = n, where p is a prime factor of m ( m runs from 1 to n-1 ), then a(n) = Sum_{m} a(m).
The number of standard tableaux of the integer partition with Heinz number n (for the definition of the Heinz number of a partition see the next comment). The proof follows from Lemma 2.8.2 of the Sagan reference. Examples: (i) a(6)=2; indeed 6 = 2*3 is the Heinz number of the partition [1,2] and, obviously, the Ferrers board admits 2 standard tableaux; (ii) a(60)=35; indeed, 60 = 2*2*3*5 is the Heinz number of the partition [1,1,2,3] and the hook-lengths of its Ferrer board are 6,3,1,4,1,2,1; then, the number of standard tableaux is 7!/(6*3*4*2) = 35. - Emeric Deutsch, May 24 2015
The Heinz number of a partition p = [p_1, p_2, ..., p_r] is defined as Product(p_j-th prime, j=1...r) (concept used by Alois P. Heinz in A215366 as an "encoding" of a partition; for example, for the partition [1,1,2,4,10] the Heinz number is 2*2*3*7*29 = 2436). - Emeric Deutsch, May 24 2015
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REFERENCES
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B. E. Sagan, The Symmetric Group, Springer, 2001, New York.
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LINKS
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EXAMPLE
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For n=10; (m=5; 2*5 = 10), (m=6; (6/3)*nextprime(3) = 10), hence a(10) = a(5) + a(6) = 3.
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MAPLE
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with(numtheory):
a:= proc(n) option remember; `if`(n=1, 1,
add(a(n/q*`if`(q=2, 1, prevprime(q))), q=factorset(n)))
end:
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MATHEMATICA
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a[n_] := a[n] = If[n == 1, 1, Sum[a[n/q*If[q == 2, 1, NextPrime[q, -1]]], {q, FactorInteger[n][[All, 1]]}]]; Table[a[n], {n, 1, 100}] (* Jean-François Alcover, Oct 04 2016, after Alois P. Heinz *)
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CROSSREFS
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KEYWORD
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easy,nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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A004070
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Table of Whitney numbers W(n,k) read by antidiagonals, where W(n,k) is maximal number of pieces into which n-space is sliced by k hyperplanes, n >= 0, k >= 0.
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+30
26
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1, 1, 1, 1, 2, 1, 1, 2, 3, 1, 1, 2, 4, 4, 1, 1, 2, 4, 7, 5, 1, 1, 2, 4, 8, 11, 6, 1, 1, 2, 4, 8, 15, 16, 7, 1, 1, 2, 4, 8, 16, 26, 22, 8, 1, 1, 2, 4, 8, 16, 31, 42, 29, 9, 1, 1, 2, 4, 8, 16, 32, 57, 64, 37, 10, 1, 1, 2, 4, 8, 16, 32, 63, 99, 93, 46, 11, 1, 1, 2, 4, 8, 16, 32, 64, 120, 163
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OFFSET
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0,5
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COMMENTS
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As a number triangle, this is given by T(n,k)=sum{j=0..n, C(n,j)(-1)^(n-j)sum{i=0..j, C(j+k,i-k)}}. - Paul Barry, Aug 23 2004
As a number triangle, this is the Riordan array (1/(1-x), x(1+x)) with T(n,k)=sum{i=0..n, binomial(k,i-k)}. Diagonal sums are then A023434(n+1). - Paul Barry, Feb 16 2005
Square array A026729 -> Partial sums across rows
1 0 0 0 0 0 0 . . . . 1 1 1 1 1 1 1 . . . . . .
1 1 0 0 0 0 0 . . . . 1 2 2 2 2 2 2 . . . . . .
1 2 1 0 0 0 0 . . . . 1 3 4 4 4 4 4 . . . . . .
1 3 3 1 0 0 0 . . . . 1 4 7 8 8 8 8 . . . . . .
For other Whitney numbers see A007799.
W(n,k) is the number of length k binary sequences containing no more than n 1's. - Geoffrey Critzer, Mar 15 2010
Viewed as a number triangle, T(n,k) is the number of internal nodes of the Fibonacci tree of order n+2 at level k. A Fibonacci tree of order n (n>=2) is a complete binary tree whose left subtree is the Fibonacci tree of order n-1 and whose right subtree is the Fibonacci tree of order n-2; each of the Fibonacci trees of order 0 and 1 is defined as a single node.
(End)
Named after the American mathematician Hassler Whitney (1907-1989). - Amiram Eldar, Jun 13 2021
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REFERENCES
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Donald E. Knuth, The Art of Computer Programming, Vol. 3, 2nd edition, Addison-Wesley, Reading, MA, 1998, p. 417. [Emeric Deutsch, Jun 15 2010]
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LINKS
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Gustav Burosch, Hans-Dietrich O.F. Gronau, Jean-Marie Laborde and Ingo Warnke, On posets of m-ary words, Discrete Math., Vol. 152, No. 1-3 (1996), pp. 69-91. MR1388633 (97e:06002)
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FORMULA
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W(n, k) = if k=0 or n=0 then 1 else W(n, k-1)+W(n-1, k-1). - David Broadhurst, Jan 05 2000
E.g.f. for row n: (1 + x + x^2/2! + ... + x^n/n!)* exp(x). - Geoffrey Critzer, Mar 15 2010
G.f.: 1 / (1 - x - x*y*(1 - x^2)) = Sum_{0 <= k <= n} x^n * y^k * T(n, k). - Michael Somos, May 31 2016
T(n, 0) = T(n, n) = 1 for n >= 0; T(n, k) = T(n-1, k-1) + T(n-2, k-1) for k=1, 2, ..., n-1, n >= 2.
T(n, k) = Sum_{m=0..n-k} binomial(k, m).
T(n,k) = 2^k for 0 <= k <= floor(n/2). (End)
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EXAMPLE
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Table W(n,k) begins:
1 1 1 1 1 1 1 ...
1 2 3 4 5 6 7 ...
1 2 4 7 11 16 22 ...
W(2,4) = 11 because there are 11 length 4 binary sequences containing no more than 2 1's: {0, 0, 0, 0}, {0, 0, 0, 1}, {0, 0, 1, 0}, {0, 0, 1, 1}, {0, 1, 0, 0}, {0, 1, 0, 1}, {0, 1, 1, 0}, {1, 0, 0, 0}, {1, 0, 0, 1}, {1, 0, 1, 0}, {1, 1, 0, 0}. - Geoffrey Critzer, Mar 15 2010
Table T(n, k) begins:
1
1 1
1 2 1
1 2 3 1
1 2 4 4 1
1 2 4 7 5 1
1 2 4 8 11 6 1
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MATHEMATICA
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Transpose[ Table[Table[Sum[Binomial[n, k], {k, 0, m}], {m, 0, 15}], {n, 0, 15}]] // Grid (* Geoffrey Critzer, Mar 15 2010 *)
T[ n_, k_] := Sum[ Binomial[n, j] (-1)^(n - j) Sum[ Binomial[j + k, i - k], {i, 0, j}], {j, 0, n}]; (* Michael Somos, May 31 2016 *)
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PROG
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(PARI) /* array read by antidiagonals up coordinate index functions */
t1(n) = binomial(floor(3/2 + sqrt(2+2*n)), 2) - (n+1); /* A025581 */
t2(n) = n - binomial(floor(1/2 + sqrt(2+2*n)), 2); /* A002262 */
/* define the sequence array function for A004070 */
W(n, k) = sum(i=0, n, binomial(k, i));
/* visual check ( origin 0, 0 ) */
printp(matrix(7, 7, n, k, W(n-1, k-1)));
/* print the sequence entries by antidiagonals going up ( origin 0, 0 ) */
print1("S A004070 "); for(n=0, 32, print1(W(t1(n), t2(n))", "));
print1("T A004070 "); for(n=33, 61, print1(W(t1(n), t2(n))", "));
(PARI) T(n, k)=sum(m=0, n-k, binomial(k, m)) \\ Jianing Song, May 30 2022
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CROSSREFS
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Rows are: A000012, A000027, A000124, A000125, A000127, A006261, A008859, A008860, A008861, A008862, A008863. - Geoffrey Critzer, Mar 15 2010
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KEYWORD
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AUTHOR
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EXTENSIONS
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More terms from Larry Reeves (larryr(AT)acm.org), Mar 20 2000
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STATUS
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approved
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A334997
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Array T read by ascending antidiagonals: T(n, k) = Sum_{d divides n} T(d, k-1) with T(n, 0) = 1.
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+30
25
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1, 1, 1, 1, 2, 1, 1, 2, 3, 1, 1, 3, 3, 4, 1, 1, 2, 6, 4, 5, 1, 1, 4, 3, 10, 5, 6, 1, 1, 2, 9, 4, 15, 6, 7, 1, 1, 4, 3, 16, 5, 21, 7, 8, 1, 1, 3, 10, 4, 25, 6, 28, 8, 9, 1, 1, 4, 6, 20, 5, 36, 7, 36, 9, 10, 1, 1, 2, 9, 10, 35, 6, 49, 8, 45, 10, 11, 1, 1, 6, 3, 16, 15, 56, 7, 64, 9, 55, 11, 12, 1
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OFFSET
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1,5
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COMMENTS
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T(n, k) is called the generalized divisor function (see Beekman).
As an array with offset n=1, k=0, T(n,k) is the number of length-k chains of divisors of n. For example, the T(4,3) = 10 chains are: 111, 211, 221, 222, 411, 421, 422, 441, 442, 444. - Gus Wiseman, Aug 04 2022
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REFERENCES
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Richard Beekman, An Introduction to Number-Theoretic Combinatorics, Lulu Press 2017.
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LINKS
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FORMULA
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T(n, k) = Sum_{d divides n} T(d, k-1) with T(n, 0) = 1 (see Theorem 3 in Beekman's article).
T(i*j, k) = T(i, k)*T(j, k) if i and j are coprime positive integers (see Lemma 1 in Beekman's article).
T(p^m, k) = binomial(m+k, k) for every prime p (see Lemma 2 in Beekman's article).
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EXAMPLE
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Array begins:
k=0 k=1 k=2 k=3 k=4 k=5 k=6 k=7 k=8
n=1: 1 1 1 1 1 1 1 1 1
n=2: 1 2 3 4 5 6 7 8 9
n=3: 1 2 3 4 5 6 7 8 9
n=4: 1 3 6 10 15 21 28 36 45
n=5: 1 2 3 4 5 6 7 8 9
n=6: 1 4 9 16 25 36 49 64 81
n=7: 1 2 3 4 5 6 7 8 9
n=8: 1 4 10 20 35 56 84 120 165
The T(4,5) = 21 chains:
(1,1,1,1,1) (4,2,1,1,1) (4,4,2,2,2)
(2,1,1,1,1) (4,2,2,1,1) (4,4,4,1,1)
(2,2,1,1,1) (4,2,2,2,1) (4,4,4,2,1)
(2,2,2,1,1) (4,2,2,2,2) (4,4,4,2,2)
(2,2,2,2,1) (4,4,1,1,1) (4,4,4,4,1)
(2,2,2,2,2) (4,4,2,1,1) (4,4,4,4,2)
(4,1,1,1,1) (4,4,2,2,1) (4,4,4,4,4)
The T(6,3) = 16 chains:
(1,1,1) (3,1,1) (6,2,1) (6,6,1)
(2,1,1) (3,3,1) (6,2,2) (6,6,2)
(2,2,1) (3,3,3) (6,3,1) (6,6,3)
(2,2,2) (6,1,1) (6,3,3) (6,6,6)
The triangular form T(n-k,k) gives the number of length k chains of divisors of n - k. It begins:
1
1 1
1 2 1
1 2 3 1
1 3 3 4 1
1 2 6 4 5 1
1 4 3 10 5 6 1
1 2 9 4 15 6 7 1
1 4 3 16 5 21 7 8 1
1 3 10 4 25 6 28 8 9 1
1 4 6 20 5 36 7 36 9 10 1
1 2 9 10 35 6 49 8 45 10 11 1
(End)
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MATHEMATICA
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T[n_, k_]:=If[n==1, 1, Product[Binomial[Extract[Extract[FactorInteger[n], i], 2]+k, k], {i, 1, Length[FactorInteger[n]]}]]; Table[T[n-k, k], {n, 1, 13}, {k, 0, n-1}]//Flatten
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PROG
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(PARI) T(n, k) = if (k==0, 1, sumdiv(n, d, T(d, k-1)));
matrix(10, 10, n, k, T(n, k-1)) \\ to see the array for n>=1, k >=0; \\ Michel Marcus, May 20 2020
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CROSSREFS
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Cf. A000217 (4th row), A000290 (6th row), A000292 (8th row), A000332 (16th row), A000389 (32nd row), A000537 (36th row), A000578 (30th row), A002411 (12th row), A002417 (24th row), A007318, A027800 (48th row), A335078, A335079.
Column k = 2 of the array is A007425.
Column k = 3 of the array is A007426.
Column k = 4 of the array is A061200.
The transpose of the array is A077592.
The subdiagonal n = k + 1 of the array is A163767.
The version counting all multisets of divisors (not just chains) is A343658.
Diagonal n = k of the array is A343939.
Antidiagonal sums of the array (or row sums of the triangle) are A343940.
A067824(n) counts strict chains of divisors starting with n.
A074206(n) counts strict chains of divisors from n to 1.
A251683(n,k) counts strict length k + 1 chains of divisors from n to 1.
A253249(n) counts nonempty chains of divisors of n.
A334996(n,k) counts strict length k chains of divisors from n to 1.
A337255(n,k) counts strict length k chains of divisors starting with n.
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KEYWORD
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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A048887
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Array T read by antidiagonals, where T(m,n) = number of compositions of n into parts <= m.
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+30
19
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1, 1, 1, 1, 2, 1, 1, 2, 3, 1, 1, 2, 4, 5, 1, 1, 2, 4, 7, 8, 1, 1, 2, 4, 8, 13, 13, 1, 1, 2, 4, 8, 15, 24, 21, 1, 1, 2, 4, 8, 16, 29, 44, 34, 1, 1, 2, 4, 8, 16, 31, 56, 81, 55, 1, 1, 2, 4, 8, 16, 32, 61, 108, 149, 89, 1, 1, 2, 4, 8, 16, 32, 63, 120, 208, 274, 144, 1
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OFFSET
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1,5
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COMMENTS
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Taking finite differences of array columns from the top down, we obtain (1; 1,1; 1,2,1; 1,4,2,1; ...) = A048004 rows. - Gary W. Adamson, Aug 20 2010
T(m,n) is the number of binary words of length n-1 with < m consecutive 1's. - Geoffrey Critzer, Sep 02 2012
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REFERENCES
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J. Riordan, An Introduction to Combinatorial Analysis, Princeton University Press, Princeton, NJ, 1978, p. 154.
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LINKS
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FORMULA
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G.f.: (1-z)/[1-2z+z^(t+1)].
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EXAMPLE
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T(2,5) counts 11111, 1112, 1121, 1211, 2111, 122, 212, 221, where "1211" abbreviates the composition 1+2+1+1.
These eight compositions correspond respectively to: {0,0,0,0}, {0,0,0,1}, {0,0,1,0}, {0,1,0,0}, {1,0,0,0}, {0,1,0,1}, {1,0,0,1}, {1,0,1,0} per the bijection given by N. J. A. Sloane in A048004. - Geoffrey Critzer, Sep 02 2012
The array begins:
1, 1, 1, 1, 1, 1, 1, ...
1, 2, 3, 5, 8, 13, ...
1, 2, 4, 7, 13, ...
1, 2, 4, 8, ...
1, 2, 4, ...
1, 2, ...
1, ...
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MAPLE
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G := t->(1-z)/(1-2*z+z^(t+1)): T := (m, n)->coeff(series(G(m), z=0, 30), z^n): matrix(7, 12, T);
# second Maple program:
T:= proc(m, n) option remember; `if`(n=0 or m=1, 1,
add(T(m, n-j), j=1..min(n, m)))
end:
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MATHEMATICA
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Table[nn=10; a=(1-x^k)/(1-x); b=1/(1-x); c=(1-x^(k-1))/(1-x); CoefficientList[ Series[a b/(1-x^2 b c), {x, 0, nn}], x], {k, 1, nn}]//Grid (* Geoffrey Critzer, Sep 02 2012 *)
T[m_, n_] := T[m, n] = If[n == 0 || m == 1, 1, Sum[T[m, n-j], {j, 1, Min[n, m]}]]; Table[Table[T[1+d-n, n], {n, 1, d}], {d, 1, 14}] // Flatten (* Jean-François Alcover, Nov 12 2014, after Alois P. Heinz *)
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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A208447
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Sum of the k-th powers of the numbers of standard Young tableaux over all partitions of n; square array A(n,k), n>=0, k>=0, read by antidiagonals.
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+30
18
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1, 1, 1, 1, 1, 2, 1, 1, 2, 3, 1, 1, 2, 4, 5, 1, 1, 2, 6, 10, 7, 1, 1, 2, 10, 24, 26, 11, 1, 1, 2, 18, 64, 120, 76, 15, 1, 1, 2, 34, 180, 596, 720, 232, 22, 1, 1, 2, 66, 520, 3060, 8056, 5040, 764, 30, 1, 1, 2, 130, 1524, 16076, 101160, 130432, 40320, 2620, 42
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OFFSET
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0,6
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LINKS
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EXAMPLE
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A(3,2) = 1^2 + 2^2 + 1^2 = 6 = 3! because 3 has partitions 111, 21, 3 with 1, 2, 1 standard Young tableaux, respectively:
.111. . 21 . . . . . . . . 3 . . . .
+---+ +------+ +------+ +---------+
| 1 | | 1 2 | | 1 3 | | 1 2 3 |
| 2 | | 3 .--+ | 2 .--+ +---------+
| 3 | +---+ +---+
+---+
Square array A(n,k) begins:
1, 1, 1, 1, 1, 1, 1, ...
1, 1, 1, 1, 1, 1, 1, ...
2, 2, 2, 2, 2, 2, 2, ...
3, 4, 6, 10, 18, 34, 66, ...
5, 10, 24, 64, 180, 520, 1524, ...
7, 26, 120, 596, 3060, 16076, 86100, ...
11, 76, 720, 8056, 101160, 1379176, 19902600, ...
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MAPLE
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h:= proc(l) local n; n:=nops(l); add(i, i=l)! /mul(mul(1+l[i]-j
+add(`if`(l[k]>=j, 1, 0), k=i+1..n), j=1..l[i]), i=1..n)
end:
g:= proc(n, i, k, l) `if`(n=0, h(l)^k, `if`(i<1, 0, g(n, i-1, k, l)
+ `if`(i>n, 0, g(n-i, i, k, [l[], i]))))
end:
A:= (n, k)-> `if`(n=0, 1, g(n, n, k, [])):
seq(seq(A(n, d-n), n=0..d), d=0..10);
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MATHEMATICA
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h[l_] := With[{n = Length[l]}, Sum[i, {i, l}]! / Product[Product[1 + l[[i]] - j + Sum [If[l[[k]] >= j, 1, 0], { k, i+1, n}], {j, 1, l[[i]]}], {i, 1, n}]]; g[n_, i_, k_, l_] := If[n == 0, h[l]^k, If[i < 1, 0, g[n, i-1, k, l] + If[i > n, 0, g[n-i, i, k, Append[l, i]]]]]; a [n_, k_] := If[n == 0, 1, g[n, n, k, {}]]; Table[Table[a[n, d-n], {n, 0, d}], {d, 0, 10}] // Flatten (* Jean-François Alcover, Dec 11 2013, translated from Maple *)
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CROSSREFS
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Columns 0-10 give: A000041, A000085, A000142, A130721, A129627, A218432, A218433, A218434, A218435, A218436, A218437.
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KEYWORD
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AUTHOR
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STATUS
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approved
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A104763
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Triangle read by rows: Fibonacci(1), Fibonacci(2), ..., Fibonacci(n) in row n.
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+30
14
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1, 1, 1, 1, 1, 2, 1, 1, 2, 3, 1, 1, 2, 3, 5, 1, 1, 2, 3, 5, 8, 1, 1, 2, 3, 5, 8, 13, 1, 1, 2, 3, 5, 8, 13, 21, 1, 1, 2, 3, 5, 8, 13, 21, 34, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233
(list;
table;
graph;
refs;
listen;
history;
text;
internal format)
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OFFSET
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1,6
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COMMENTS
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Triangle of A104762, Fibonacci sequence in each row starts from the right.
The triangle or chess sums, see A180662 for their definitions, link the Fibonacci(n) triangle to sixteen different sequences, see the crossrefs. The knight sums Kn14 - Kn18 have been added. As could be expected all sums are related to the Fibonacci numbers. - Johannes W. Meijer, Sep 22 2010
Sequence B is called a reluctant sequence of sequence A, if B is triangle array read by rows: row number k coincides with first k elements of the sequence A. Sequence A104763 is reluctant sequence of Fibonacci numbers (A000045), except 0. - Boris Putievskiy, Dec 13 2012
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LINKS
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FORMULA
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F(1) through F(n) starting from the left in n-th row.
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EXAMPLE
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First few rows of the triangle are:
1;
1, 1;
1, 1, 2;
1, 1, 2, 3;
1, 1, 2, 3, 5;
1, 1, 2, 3, 5, 8;
1, 1, 2, 3, 5, 8, 13; ...
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MATHEMATICA
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Table[Fibonacci[k], {n, 15}, {k, n}]//Flatten (* G. C. Greubel, Jul 13 2019 *)
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PROG
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(Haskell)
a104763 n k = a104763_tabl !! (n-1) !! (k-1)
a104763_row n = a104763_tabl !! (n-1)
a104763_tabl = map (flip take $ tail a000045_list) [1..]
(PARI) for(n=1, 15, for(k=1, n, print1(fibonacci(k), ", "))) \\ G. C. Greubel, Jul 13 2019
(Magma) [Fibonacci(k): k in [1..n], n in [1..15]]; // G. C. Greubel, Jul 13 2019
(Sage) [[fibonacci(k) for k in (1..n)] for n in (1..15)] # G. C. Greubel, Jul 13 2019
(GAP) Flat(List([1..15], n-> List([1..n], Fibonacci(k) ))) # G. C. Greubel, Jul 13 2019
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CROSSREFS
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KEYWORD
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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A144328
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A002260 preceded by a column of 1's: a (1, 1, 2, 3, 4, 5, ...) crescendo triangle by rows.
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+30
14
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1, 1, 1, 1, 1, 2, 1, 1, 2, 3, 1, 1, 2, 3, 4, 1, 1, 2, 3, 4, 5, 1, 1, 2, 3, 4, 5, 6, 1, 1, 2, 3, 4, 5, 6, 7, 1, 1, 2, 3, 4, 5, 6, 7, 8, 1, 1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 1, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11
(list;
table;
graph;
refs;
listen;
history;
text;
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OFFSET
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1,6
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COMMENTS
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Eigensequence of the triangle = A000142, the factorials.
The triangle as an infinite lower triangular matrix * [1,2,3,...] = A064999.
Generated from A128227 by rotating each row by one position to the right. - R. J. Mathar, Sep 25 2008
A sequence B is called a reluctant sequence of sequence A, if B is triangle array read by rows: row number k coincides with first k elements of the sequence A. Sequence A144328 is the reluctant sequence of A028310 (1 followed by the natural numbers). - Boris Putievskiy, Dec 12 2012
If offset were changed to 0, a(n) would equal the
Let S_n be the set of partitions of n into distinct parts where the number of parts is maximal for that n. For example, for n=6, the set S_6 consists of just one such partition: S_6={1,2,3}. Similarly, for n=7, S_7={1,2,4}, But for n=8, S_8 will contain two partitions S_8= { {1,2,5}, {1,3,4} }. Then |S(n)| = a(n+1). Cf. A178702. - David S. Newman and Benoit Jubin, Dec 13 2010
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LINKS
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FORMULA
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Triangle A002260 (natural numbers crescendo triangle) preceded by a column of 1's, = a (1, 1, 2, 3, 4, 5, ...) crescendo triangle by rows.
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EXAMPLE
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First few rows of the triangle:
1;
1, 1;
1, 1, 2;
1, 1, 2, 3;
1, 1, 2, 3, 4;
1, 1, 2, 3, 4, 5;
...
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MATHEMATICA
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Flatten[Table[Join[{1}, Range[n]], {n, 0, 11}]] (* Harvey P. Dale, Aug 10 2013 *)
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PROG
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(Haskell)
a144328 n k = a144328_tabl !! (n-1) !! (k-1)
a144328_row n = a144328_tabl !! (n-1)
a144328_tabl = [1] : map (\xs@(x:_) -> x : xs) a002260_tabl
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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A152977
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Square array A(n,k), n>=0, k>=0, read by antidiagonals: A(n,k) is the number of partitions of 2^n into powers of 2 less than or equal to 2^k.
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+30
13
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1, 1, 1, 1, 2, 1, 1, 2, 3, 1, 1, 2, 4, 5, 1, 1, 2, 4, 9, 9, 1, 1, 2, 4, 10, 25, 17, 1, 1, 2, 4, 10, 35, 81, 33, 1, 1, 2, 4, 10, 36, 165, 289, 65, 1, 1, 2, 4, 10, 36, 201, 969, 1089, 129, 1, 1, 2, 4, 10, 36, 202, 1625, 6545, 4225, 257, 1, 1, 2, 4, 10, 36, 202, 1827, 17361, 47905, 16641, 513, 1
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OFFSET
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0,5
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COMMENTS
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Column sequences converge towards A002577.
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LINKS
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FORMULA
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A(n,k) = [x^2^(n-1)] 1/(1-x) * 1/Product_{j=0..k-1} (1-x^(2^j)) for n>0; A(0,k) = 1.
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EXAMPLE
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A(3,2) = 9, because there are 9 partitions of 2^3=8 into powers of 2 less than or equal to 2^2=4: [4,4], [4,2,2], [4,2,1,1], [4,1,1,1,1], [2,2,2,2], [2,2,2,1,1], [2,2,1,1,1,1], [2,1,1,1,1,1,1], [1,1,1,1,1,1,1,1].
Square array A(n,k) begins:
1, 1, 1, 1, 1, 1, ...
1, 2, 2, 2, 2, 2, ...
1, 3, 4, 4, 4, 4, ...
1, 5, 9, 10, 10, 10, ...
1, 9, 25, 35, 36, 36, ...
1, 17, 81, 165, 201, 202, ...
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MAPLE
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b:= proc(n, j) local nn, r;
if n<0 then 0
elif j=0 then 1
elif j=1 then n+1
elif n<j then b(n, j):= b(n-1, j) +b(2*n, j-1)
else nn:= 1 +floor(n);
r:= n-nn;
(nn-j) *binomial(nn, j) *add(binomial(j, h)
/(nn-j+h) *b(j-h+r, j) *(-1)^h, h=0..j-1)
fi
end:
A:= (n, k)-> `if`(n=0, 1, b(2^(n-k), k)):
seq(seq(A(n, d-n), n=0..d), d=0..11);
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MATHEMATICA
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b[n_, j_] := Module[{nn, r}, Which[n < 0, 0, j == 0, 1, j == 1, n+1, n < j, b[n, j] = b[n-1, j]+b[2*n, j-1], True, nn = 1+Floor[n]; r := n-nn; (nn-j)*Binomial[nn, j]*Sum[Binomial[j, h]/(nn-j+h)*b[j-h+r, j]*(-1)^h, {h, 0, j-1}]]]; a[n_, k_] := If[n == 0, 1, b[2^(n-k), k]]; Table[Table[a[n, d-n], {n, 0, d}], {d, 0, 11}] // Flatten (* Jean-François Alcover, Dec 18 2013, translated from Maple *)
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CROSSREFS
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Columns k=0-10 give: A000012, A094373, A028400(n-2) for n>1, A210772, A210773, A210774, A210775, A210776, A210777, A210778, A210779.
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KEYWORD
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AUTHOR
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STATUS
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approved
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A259799
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Array read by antidiagonals upwards: T(n,k) = number of partitions of k^n into n-th powers (n>=1, k>=0).
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+30
11
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1, 1, 1, 1, 1, 2, 1, 1, 2, 3, 1, 1, 2, 4, 5, 1, 1, 2, 5, 8, 7, 1, 1, 2, 7, 17, 19, 11, 1, 1, 2, 9, 36, 62, 43, 15, 1, 1, 2, 13, 88, 253, 258, 98, 22, 1, 1, 2, 19, 218, 1104, 1886, 1050, 220, 30, 1, 1, 2, 27, 550, 5082, 15772, 14800, 4365, 504, 42, 1, 1, 2, 40, 1413, 24119, 140549, 241582, 118238, 18012, 1116, 56
(list;
table;
graph;
refs;
listen;
history;
text;
internal format)
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OFFSET
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1,6
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LINKS
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EXAMPLE
|
The array begins:
1, 1, 2, 3, 5, 7, 11, 15, 22, 30, ...
1, 1, 2, 4, 8, 19, 43, 98, 220, 504, ...
1, 1, 2, 5, 17, 62, 258, 1050, 4365, 18012, ...
1, 1, 2, 7, 36, 253, 1886, 14800, 118238, ...
1, 1, 2, 9, 88, 1104, 15772, 241582, ...
...
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MAPLE
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b:= proc(n, i, k) option remember; `if`(n=0 or i=1, 1,
`if`(i=2, 1+iquo(n, i^k), b(n, i-1, k)+
`if`(i^k>n, 0, b(n-i^k, i, k))))
end:
T:= (n, k)-> b(k^n, k, n):
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MATHEMATICA
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b[n_, i_, k_] := b[n, i, k] = If[n==0 || i==1, 1, If[i==2, 1+Quotient[n, i^k], b[n, i-1, k] + If[i^k>n, 0, b[n-i^k, i, k]]]]; T[n_, k_] := b[k^n, k, n]; Table[ Table[ T[d-k, k], {k, 0, d-1}], {d, 1, 12}] // Flatten (* Jean-François Alcover, Jul 15 2015, after Alois P. Heinz *)
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CROSSREFS
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KEYWORD
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|
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AUTHOR
|
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EXTENSIONS
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STATUS
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approved
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