A351800 a(n) = [x^n] 1/Product_{j=1..n} (1 - j^3*x).

1, 1, 73, 28800, 33120201, 83648533275, 393764054984212, 3103381708489548640, 37965284782803741391413, 681476650259874114533077575, 17184647574689079046814198039765, 588057239856779143071625300022102376, 26548105106818292578525347802793561068860

Offset: 0

Author

Alois P. Heinz, Feb 19 2022

Keywords

• nonn

Examples

			a(2) = (1*1)^3 + (1*2)^3 + (2*2)^3 = 1 + 8 + 64 = 73.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..149

Crossrefs

Cf. A000442, A001700, A007820, A088218, A098436, A269948, A298851, A351804.

Programs

Maple

b:= proc(n, k) option remember; `if`(k=0, 1,
      add(b(j, k-1)*j^3, j=1..n))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..15);

Mathematica

Table[SeriesCoefficient[Product[1/(1 - k^3*x), {k, 1, n}], {x, 0, n}], {n, 0, 15}] (* Vaclav Kotesovec, May 17 2025 *)

Formula

a(n) = Sum_{p in {1..n}^n : p_i <= p_{i+1}} Product_{j=1..n} p_j^3.

a(n) = A098436(2n-1,n-1) = A269948(2n,n).

a(n) ~ c * d^n * n^(3*n - 1/2), where d = 1.54371040458513693750053812318801418996889528987425... and c = 0.71526493063554190404119140313248864511356727815244... - Vaclav Kotesovec, May 13 2025