A316152 Inverse Euler transform of n^2.

1, 3, 5, 1, -6, -17, -4, 29, 56, 7, -158, -255, 56, 878, 1234, -725, -4966, -5852, 6132, 28410, 26932, -46529, -162814, -117479, 332350, 929292, 454328, -2279218, -5259270, -1252181, 15199212, 29375985, -1279006, -99212897, -161079712, 60433632, 635914664, 860993882

Offset: 1

Author

Seiichi Manyama, Jun 25 2018

Keywords

• sign

Examples

			(1-x)^(-1)*(1-x^2)^(-3)*(1-x^3)^(-5)*(1-x^4)^(-1)*(1-x^5)^6* ... = 1 + x + 4*x^2 + 9*x^3 + 16*x^4 + 25*x^5 + ... .
		

Links

• Seiichi Manyama, Table of n, a(n) for n = 1..1000
• N. J. A. Sloane, Transforms

Crossrefs

Cf. A000290, A253909, A316150.

Programs

Maple

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(binomial(a(i)+j-1, j)*b(n-i*j, i-1), j=0..n/i)))
    end:
a:= proc(n) option remember; n^2-b(n, n-1) end:
seq(a(n), n=1..40);  # Alois P. Heinz, Jun 29 2018

Mathematica

b[n_, i_] := b[n, i] = If[n == 0, 1, If[i < 1, 0, Sum[Binomial[a[i] + j - 1, j]*b[n - i*j, i - 1], {j, 0, n/i}]]];
a[n_] := n^2 - b[n, n - 1];
a /@ Range[40] (* Jean-François Alcover, Jan 06 2020, after Alois P. Heinz *)

Formula

Product_{k>=1} (1-x^k)^(-a(k)) = 1 + Sum_{k>=1} A000290(k)*x^k.

G.f.: Sum_{k>=1} mu(k)*log(1 + x^k*(1 + x^k)/(1 - x^k)^3)/k. - Ilya Gutkovskiy, May 18 2019