A092484 Expansion of Product_{m>=1} (1 + m^2*q^m).

1, 1, 4, 13, 25, 77, 161, 393, 726, 2010, 3850, 7874, 16791, 31627, 69695, 139560, 255997, 482277, 986021, 1716430, 3544299, 6507128, 11887340, 21137849, 38636535, 70598032, 123697772, 233003286, 412142276, 711896765, 1252360770

Offset

0,3

Comments

Sum of squares of products of terms in all partitions of n into distinct parts.

Links

Alois P. Heinz, Table of n, a(n) for n = 0..10000

Formula

G.f.: exp(Sum_{k>=1} Sum_{j>=1} (-1)^(k+1)*j^(2*k)*x^(j*k)/k). - Ilya Gutkovskiy, Jun 14 2018

Conjecture: log(a(n)) ~ sqrt(2*n) * (log(2*n) - 2). - Vaclav Kotesovec, Dec 27 2020

Example

The partitions of 6 into distinct parts are 6, 1+5, 2+4, 1+2+3, the corresponding squares of products are 36, 25, 64, 36 and their sum is a(6) = 161.

Maple

b:= proc(n, i) option remember; (m->
      `if`(m<n, 0, `if`(n=m, i!^2, b(n, i-1)+
      `if`(i>n, 0, i^2*b(n-i, i-1)))))(i*(i+1)/2)
    end:
a:= n-> b(n$2):
seq(a(n), n=0..40);  # Alois P. Heinz, Sep 10 2017

Mathematica

Take[ CoefficientList[ Expand[ Product[1 + m^2*q^m, {m, 100}]], q], 31] (* Robert G. Wilson v, Apr 05 2005 *)

Prog

(PARI) N=66; x='x+O('x^N); Vec(prod(n=1, N, 1+n^2*x^n)) \\ Seiichi Manyama, Sep 10 2017

Crossrefs

Cf. A022629, A077335, A265844, A285737, A292165.

Column k=2 of A292189.

Sequence in context: A056708 A307271 A333297 * A091823 A024834 A143867

Adjacent sequences: A092481 A092482 A092483 * A092485 A092486 A092487

Keyword

nonn

Author

Jon Perry, Apr 04 2004

Extensions

More terms from Robert G. Wilson v, Apr 05 2004

Status

approved