A335633 Number of ordered ways of writing the n-th n-gonal number as a sum of n n-gonal numbers (with 0's allowed).
1, 1, 3, 6, 5, 95, 336, 2597, 26832, 197577, 1847800, 14621101, 129754956, 1146534701, 12342194879, 161225146370, 2464561564936, 39642413790129, 620059254486798, 9430493858327959, 136438759335452360, 1881721996407396801, 24999081626667425376, 321601467988647184779
Offset: 0
Keywords
Examples
a(3) = 6 because the third triangular number is 6 and we have [6, 0, 0], [0, 6, 0], [0, 0, 6], [3, 3, 0], [3, 0, 3] and [0, 3, 3].
Links
- Eric Weisstein's World of Mathematics, Polygonal Number
- Index to sequences related to polygonal numbers
Programs
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Mathematica
Table[SeriesCoefficient[Sum[x^(k (k (n - 2) - n + 4)/2), {k, 0, n}]^n, {x, 0, n (n^2 - 3 n + 4)/2}], {n, 0, 23}] -
PARI
p(n,k) = {k * (k * (n - 2) - n + 4) / 2} a(n) = {my(m=p(n,n)); polcoef((sum(k=0, n, x^p(n,k)) + O(x*x^m))^n, m)} \\ Andrew Howroyd, Oct 03 2020
Formula
a(n) = [x^p(n,n)] (Sum_{k=0..n} x^p(n,k))^n, where p(n,k) = k * (k * (n - 2) - n + 4) / 2 is the k-th n-gonal number.