A292165 -id:A292165 - cp's OEIS frontend

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

Jon Perry, Apr 04 2004

nonn

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

			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.

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

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

Column k=2 of A292189.

b:= proc(n, i) option remember; (m->
      `if`(mn, 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

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

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

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

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

A292068 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of Product_{j>=1} 1/(1 + j^k*x^j).

Original entry on oeis.org

1, 1, -1, 1, -1, 0, 1, -1, -1, -1, 1, -1, -3, -2, 1, 1, -1, -7, -6, 2, -1, 1, -1, -15, -20, 6, -1, 1, 1, -1, -31, -66, 20, 5, 4, -1, 1, -1, -63, -212, 66, 71, 40, -1, 2, 1, -1, -127, -666, 212, 605, 442, 11, 18, -2, 1, -1, -255, -2060, 666, 4439, 4660, 215, 226, -22, 2

Offset: 0

Seiichi Manyama, Sep 12 2017

			Square array begins:
    1,  1,  1,   1,   1, ...
   -1, -1, -1,  -1,  -1, ...
    0, -1, -3,  -7, -15, ...
   -1, -2, -6, -20, -66, ...
    1,  2,  6,  20,  66, ...

Columns k=0..2 give A081362, A022693, A292165.
Rows n=0..2 give A000012, (-1)*A000012, (-1)*A000225.
Main diagonal gives A292072.
Cf. A292189, A292193.

b:= proc(n, i, k) option remember; (m->
      `if`(mn, 0, i^k*b(n-i, i-1, k)))))(i*(i+1)/2)
    end:
A:= proc(n, k) option remember; `if`(n=0, 1,
      -add(b(n-i$2, k)*A(i, k), i=0..n-1))
    end:
seq(seq(A(n, d-n), n=0..d), d=0..12);  # Alois P. Heinz, Sep 12 2017

b[n_, i_, k_] := b[n, i, k] = If[# < n, 0, If[n == #, i!^k, b[n, i-1, k] + If[i > n, 0, i^k b[n-i, i-1, k]]]]&[i(i+1)/2];
A[n_, k_] := A[n, k] = If[n == 0, 1, -Sum[b[n-i, n-i, k] A[i, k], {i, 0, n-1}]];
Table[A[n, d-n], {d, 0, 12}, {n, 0, d}] // Flatten (* Jean-François Alcover, Nov 20 2019, after Alois P. Heinz *)

from sympy.core.cache import cacheit
from sympy import factorial as f
@cacheit
def b(n, i, k):
    m=i*(i + 1)/2
    return 0 if mn else i**k*b(n - i, i - 1, k))
@cacheit
def A(n, k): return 1 if n==0 else -sum([b(n - i, n - i, k)*A(i, k) for i in range(n)])
for d in range(13): print([A(n, d - n) for n in range(d + 1)]) # Indranil Ghosh, Sep 14 2017, after Maple program

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

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