A292165 Expansion of Product_{k>=1} 1/(1 + k^2*x^k).

1, -1, -3, -6, 6, 5, 40, 11, 226, -516, -186, -844, 3731, -3734, 814, -33819, 85660, -46022, 210342, -411678, 593996, -2980156, 2076721, -3445584, 40785410, -37503158, 98085, -271846888, 336918770, -295108832, 2178341296, -2404059340, 6127604258

Offset: 0

Author

Seiichi Manyama, Sep 10 2017

Keywords

• sign

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..3145

Crossrefs

Cf. A077335, A092484.

Programs

Maple

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

Mathematica

b[n_, i_] := b[n, i] = Function[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];
a[n_] := a[n] = If[n == 0, 1, -Sum[b[n - i, n - i]*a[i], {i, 0, n - 1}]];
Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Mar 21 2022, after Alois P. Heinz *)

PARI

N=66; x='x+O('x^N); Vec(1/prod(n=1, N, 1+n^2*x^n))

Formula

Convolution inverse of A092484.

From Vaclav Kotesovec, Sep 10 2017: (Start)

a(n) ~ (-1)^n * c * 3^(2*n/3), where

c = 0.717271758899891528435966115495396784611147877234945... if mod(n,3)=0

c = 0.387695187106751505296020614217498222070185848125472... if mod(n,3)=1

c = 0.241939482775588594057384356004734639024152664456553... if mod(n,3)=2

(End)

G.f.: exp(Sum_{k>=1} Sum_{j>=1} (-1)^k*j^(2*k)*x^(j*k)/k). - Ilya Gutkovskiy, Jun 18 2018