This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A320591 #30 Feb 07 2025 21:32:24
%S A320591 1,1,0,1,-2,4,-7,11,-16,23,-36,65,-129,256,-473,772,-1028,835,776,
%T A320591 -5755,17562,-41750,86678,-165145,299949,-541837,1020029,-2068203,
%U A320591 4509512,-10252952,23465297,-52762788,115160832,-243018459,496094524,-982431070,1894710043,-3574095362
%N A320591 Expansion of Product_{k>=1} (1 + x^k/(1 + x)^k).
%C A320591 After the first term, this is the second term of the n-th differences of A000009, or column n=1 of A378622. - _Gus Wiseman_, Feb 03 2025
%H A320591 G. C. Greubel, <a href="/A320591/b320591.txt">Table of n, a(n) for n = 0..1000</a>
%F A320591 G.f.: exp(Sum_{k>=1} (-1)^(k+1)*x^k/(k*((1 + x)^k - x^k))).
%F A320591 G.f.: exp(Sum_{k>=1} A000593(k)*x^k/(k*(1 + x)^k)).
%F A320591 From _Peter Bala_, Dec 22 2020: (Start)
%F A320591 O.g.f.: Sum_{n >= 0} x^(n*(n+1)/2)/Product_{k = 1..n} ((1 + x)^k - x^k). Cf. A307548.
%F A320591 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)
%F A320591 a(n+1) = Sum_{k=0..n} (-1)^(n-k) binomial(n,k) A000009(k+1). - _Gus Wiseman_, Feb 03 2025
%p A320591 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 A320591 nmax = 37; CoefficientList[Series[Product[(1 + x^k/(1 + x)^k), {k, 1, nmax}], {x, 0, nmax}], x]
%t A320591 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]
%t A320591 Prepend[Table[Differences[PartitionsQ/@Range[0,k+1],k][[2]],{k,0,30}],1] (* _Gus Wiseman_, Jan 29 2025 *)
%o A320591 (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 A320591 (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 A320591 Cf. A000593, A320589, A307548.
%Y A320591 The version for non-strict partitions is A320590, row n=1 of A175804.
%Y A320591 Column n=1 (except first term) of A378622. See also A293467, A377285, A378970, A378971, A380412 (column n=0).
%Y A320591 A000009 counts strict integer partitions, differences A087897, A378972.
%Y A320591 A266232 gives zero-based binomial transform of strict partitions, differences A129519.
%Y A320591 Cf. A000041, A007442, A002865, A008284, A095195, A103446, A116608, A218482, A281425, A320568.
%K A320591 sign,easy
%O A320591 0,5
%A A320591 _Ilya Gutkovskiy_, Oct 16 2018