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A292541 a(n) is n-th term of the Euler transform of -n,1,1,1,... .

%I A292541 #11 May 07 2018 18:00:03
%S A292541 1,-1,2,-3,5,-9,18,-39,88,-200,449,-988,2131,-4527,9540,-20090,42510,
%T A292541 -90596,194299,-418105,899493,-1929000,4116944,-8742002,18484225,
%U A292541 -38974978,82086786,-172927251,364700265,-770223900,1628602725,-3445907334,7291399538
%N A292541 a(n) is n-th term of the Euler transform of -n,1,1,1,... .
%H A292541 Alois P. Heinz, <a href="/A292541/b292541.txt">Table of n, a(n) for n = 0..3260</a>
%F A292541 a(n) = [x^n] (1-x)^n / Product_{j=2..n} (1-x^j).
%F A292541 a(n) ~ (-1)^n * exp(Pi*sqrt(n/3)/2 + Pi^2/96) * 2^(n - 1/2) / (sqrt(3)*n). - _Vaclav Kotesovec_, May 07 2018
%p A292541 b:= proc(n, i, k) option remember; `if`(n=0 or i<2,
%p A292541       binomial(k+n-1, n), add(b(n-i*j, i-1, k), j=0..n/i))
%p A292541     end:
%p A292541 a:= n-> b(n$2, -n):
%p A292541 seq(a(n), n=0..35);
%p A292541 # second Maple program:
%p A292541 b:= proc(n, k) option remember; `if`(n=0, 1, add(
%p A292541       (numtheory[sigma](j)+k-1)*b(n-j, k), j=1..n)/n)
%p A292541     end:
%p A292541 a:= n-> b(n, -n):
%p A292541 seq(a(n), n=0..35);
%p A292541 # third Maple program:
%p A292541 b:= proc(n, k) option remember; `if`(n=0, 1, `if`(k=1,
%p A292541       combinat[numbpart](n), b(n, k+1)-b(n-1, k+1)))
%p A292541     end:
%p A292541 a:= n-> b(n, -n):
%p A292541 seq(a(n), n=0..35);
%t A292541 Table[SeriesCoefficient[(1 - x)^n*Product[1/(1 - x^k), {k, 2, n}], {x, 0, n}], {n, 0, 30}] (* _Vaclav Kotesovec_, May 07 2018 *)
%t A292541 b[n_, k_] := b[n, k] = If[n == 0, 1, If[k == 1, PartitionsP[n], b[n, k + 1] - b[n - 1, k + 1]]]; Table[b[n, -n], {n, 0, 40}] (* _Vaclav Kotesovec_, May 07 2018, after _Alois P. Heinz_ *)
%Y A292541 Cf. A292463.
%K A292541 sign
%O A292541 0,3
%A A292541 _Alois P. Heinz_, Sep 18 2017