A337762 Number of partitions of the n-th n-gonal number into n-gonal numbers.

1, 1, 2, 4, 8, 21, 56, 144, 370, 926, 2275, 5482, 12966, 30124, 68838, 154934, 343756, 752689, 1627701, 3479226, 7355608, 15390682, 31889732, 65465473, 133212912, 268811363, 538119723, 1069051243, 2108416588, 4129355331, 8033439333

Offset: 0

Author

Ilya Gutkovskiy, Sep 19 2020

Keywords

• nonn

Examples

			a(3) = 4 because the third triangular number is 6 and we have [6], [3, 3], [3, 1, 1, 1] and [1, 1, 1, 1, 1, 1].
		

Links

• David A. Corneth, Table of n, a(n) for n = 0..278 (first 51 terms from Vaclav Kotesovec)
• Eric Weisstein's World of Mathematics, Polygonal Number
• Index entries for sequences related to partitions
• Index to sequences related to polygonal numbers

Crossrefs

Cf. A000041, A037444, A060354, A072964, A337763, A337764.

Programs

Mathematica

nmax = 20; Table[SeriesCoefficient[Product[1/(1 - x^(k*((k*(n - 2) - n + 4)/2))), {k, 1, n}], {x, 0, n*(4 - 3*n + n^2)/2}], {n, 0, nmax}] (* Vaclav Kotesovec, Sep 19 2020 *)

Formula

a(n) = [x^p(n,n)] Product_{k=1..n} 1 / (1 - x^p(n,k)), where p(n,k) = k * (k * (n - 2) - n + 4) / 2 is the k-th n-gonal number.