A320591 - OEIS

%I #17 Sep 08 2022 08:46:23

%S 1,1,0,1,-2,4,-7,11,-16,23,-36,65,-129,256,-473,772,-1028,835,776,

%T -5755,17562,-41750,86678,-165145,299949,-541837,1020029,-2068203,

%U 4509512,-10252952,23465297,-52762788,115160832,-243018459,496094524,-982431070,1894710043,-3574095362

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

%H G. C. Greubel, <a href="/A320591/b320591.txt">Table of n, a(n) for n = 0..1000</a>

%F G.f.: exp(Sum_{k>=1} (-1)^(k+1)*x^k/(k*((1 + x)^k - x^k))).

%F G.f.: exp(Sum_{k>=1} A000593(k)*x^k/(k*(1 + x)^k)).

%F From _Peter Bala_, Dec 22 2020: (Start)

%F O.g.f.: Sum_{n >= 0}  x^(n*(n+1)/2)/Product_{k = 1..n} ((1 + x)^k - x^k). Cf. A307548.

%F Conjectural o.g.f.: (1/2) * Sum_{n >= 0} x^(n*(n-1)/2)*(1 + x)^n/( Product_{k = 1..n} ( (1 + x)^k - x^k ) ). (End)

%p seq(coeff(series(mul((1+x^k/(1+x)^k),k=1..n),x,n+1), x, n), n = 0 .. 37); # _Muniru A Asiru_, Oct 16 2018

%t nmax = 37; CoefficientList[Series[Product[(1 + x^k/(1 + x)^k), {k, 1, nmax}], {x, 0, nmax}], x]

%t nmax = 37; CoefficientList[Series[Exp[Sum[Sum[(-1)^(k/d + 1) d, {d, Divisors[k]}] x^k/(k (1 + x)^k), {k, 1, nmax}]], {x, 0, nmax}], x]

%o (PARI) m=50; x='x+O('x^m); Vec(prod(k=1, m+2, (1 + x^k/(1 + x)^k))) \\ _G. C. Greubel_, Oct 29 2018

%o (Magma) m:=50; R<x>:=PowerSeriesRing(Rationals(), m); Coefficients(R! ( (&*[(1 + x^k/(1 + x)^k): k in [1..(m+2)]]) )); // _G. C. Greubel_, Oct 29 2018

%Y Cf. A000593, A129519, A320589, A320590, A307548.

%K sign,easy

%O 0,5

%A _Ilya Gutkovskiy_, Oct 16 2018