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A316152 Inverse Euler transform of n^2.

%I A316152 #23 Jan 06 2020 09:12:36
%S A316152 1,3,5,1,-6,-17,-4,29,56,7,-158,-255,56,878,1234,-725,-4966,-5852,
%T A316152 6132,28410,26932,-46529,-162814,-117479,332350,929292,454328,
%U A316152 -2279218,-5259270,-1252181,15199212,29375985,-1279006,-99212897,-161079712,60433632,635914664,860993882
%N A316152 Inverse Euler transform of n^2.
%H A316152 Seiichi Manyama, <a href="/A316152/b316152.txt">Table of n, a(n) for n = 1..1000</a>
%H A316152 N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>
%F A316152 Product_{k>=1} (1-x^k)^(-a(k)) = 1 + Sum_{k>=1} A000290(k)*x^k.
%F A316152 G.f.: Sum_{k>=1} mu(k)*log(1 + x^k*(1 + x^k)/(1 - x^k)^3)/k. - _Ilya Gutkovskiy_, May 18 2019
%e A316152 (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 + ... .
%p A316152 b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
%p A316152       add(binomial(a(i)+j-1, j)*b(n-i*j, i-1), j=0..n/i)))
%p A316152     end:
%p A316152 a:= proc(n) option remember; n^2-b(n, n-1) end:
%p A316152 seq(a(n), n=1..40);  # _Alois P. Heinz_, Jun 29 2018
%t A316152 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}]]];
%t A316152 a[n_] := n^2 - b[n, n - 1];
%t A316152 a /@ Range[40] (* _Jean-François Alcover_, Jan 06 2020, after _Alois P. Heinz_ *)
%Y A316152 Cf. A000290, A253909, A316150.
%K A316152 sign
%O A316152 1,2
%A A316152 _Seiichi Manyama_, Jun 25 2018