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

0,3

Links

Table of n, a(n) for n=0..23.

Eric Weisstein's World of Mathematics, Polygonal Number

Index to sequences related to polygonal numbers

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.

Example

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].

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}]

Prog

(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

Crossrefs

Cf. A060354, A088218, A196010, A298329, A335634, A337762, A337763, A337764.

Sequence in context: A121867 A307132 A300673 * A009193 A144253 A152422

Adjacent sequences: A335630 A335631 A335632 * A335634 A335635 A335636

Keyword

nonn

Author

Ilya Gutkovskiy, Oct 03 2020

Status

approved