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A320590 Expansion of Product_{k>=1} 1/(1 - x^k/(1 + x)^k).

%I A320590 #20 Dec 21 2024 17:58:17
%S A320590 1,1,1,0,1,-2,5,-12,28,-63,137,-290,604,-1253,2617,-5537,11870,-25666,
%T A320590 55617,-120103,257582,-548119,1158437,-2437114,5117165,-10748530,
%U A320590 22621055,-47728657,100932549,-213750621,452855190,-958925784,2028187595,-4283531490,9033779224
%N A320590 Expansion of Product_{k>=1} 1/(1 - x^k/(1 + x)^k).
%C A320590 The zero-based binomial transform of this sequence is A000070, and if we remove first terms it becomes A000041.
%H A320590 G. C. Greubel, <a href="/A320590/b320590.txt">Table of n, a(n) for n = 0..1000</a>
%F A320590 G.f.: exp(Sum_{k>=1} x^k/(k*((1 + x)^k - x^k))).
%F A320590 G.f.: exp(Sum_{k>=1} sigma(k)*x^k/(k*(1 + x)^k)).
%p A320590 seq(coeff(series(mul(1/(1-x^k/(1+x)^k),k=1..n),x,n+1), x, n), n = 0 .. 35); # _Muniru A Asiru_, Oct 16 2018
%t A320590 nmax = 34; CoefficientList[Series[Product[1/(1 - x^k/(1 + x)^k), {k, 1, nmax}], {x, 0, nmax}], x]
%t A320590 nmax = 34; CoefficientList[Series[Exp[Sum[DivisorSigma[1, k] x^k/(k (1 + x)^k), {k, 1, nmax}]], {x, 0, nmax}], x]
%o A320590 (PARI) m=50; x='x+O('x^m); Vec(prod(k=1, m+2, 1/(1 - x^k/(1 + x)^k))) \\ _G. C. Greubel_, Oct 29 2018
%o A320590 (Magma) m:=50; R<x>:=PowerSeriesRing(Rationals(), m); Coefficients(R! ( (&*[1/(1 - x^k/(1 + x)^k): k in [1..(m+2)]]) )); // _G. C. Greubel_, Oct 29 2018
%Y A320590 Cf. A000203, A103446, A218482, A320568.
%Y A320590 Row n=1 of A175804 (except first term). Row n=0 is A281425.
%Y A320590 The version for strict partitions is A320591, row n=1 of A378622, first column A293467.
%Y A320590 A000009 counts strict integer partitions, differences A087897, A378972.
%Y A320590 A000041 counts integer partitions, differences A002865.
%Y A320590 Cf. A047966, A008284, A377056, A378621.
%K A320590 sign
%O A320590 0,6
%A A320590 _Ilya Gutkovskiy_, Oct 16 2018