A320589 -id:A320589 - cp's OEIS frontend

A320591 Expansion of Product_{k>=1} (1 + x^k/(1 + x)^k).

1, 1, 0, 1, -2, 4, -7, 11, -16, 23, -36, 65, -129, 256, -473, 772, -1028, 835, 776, -5755, 17562, -41750, 86678, -165145, 299949, -541837, 1020029, -2068203, 4509512, -10252952, 23465297, -52762788, 115160832, -243018459, 496094524, -982431070, 1894710043, -3574095362

Offset: 0

Ilya Gutkovskiy, Oct 16 2018

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

G. C. Greubel, Table of n, a(n) for n = 0..1000

Cf. A000593, A320589, A307548.
The version for non-strict partitions is A320590, row n=1 of A175804.
Column n=1 (except first term) of A378622. See also A293467, A377285, A378970, A378971, A380412 (column n=0).
A000009 counts strict integer partitions, differences A087897, A378972.
A266232 gives zero-based binomial transform of strict partitions, differences A129519.
Cf. A000041, A007442, A002865, A008284, A095195, A103446, A116608, A218482, A281425, A320568.

m:=50; R:=PowerSeriesRing(Rationals(), m); Coefficients(R! ( (&*[(1 + x^k/(1 + x)^k): k in [1..(m+2)]]) )); // G. C. Greubel, Oct 29 2018

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

nmax = 37; CoefficientList[Series[Product[(1 + x^k/(1 + x)^k), {k, 1, nmax}], {x, 0, nmax}], x]
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]
Prepend[Table[Differences[PartitionsQ/@Range[0,k+1],k][[2]],{k,0,30}],1] (* Gus Wiseman, Jan 29 2025 *)

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

G.f.: exp(Sum_{k>=1} (-1)^(k+1)*x^k/(k*((1 + x)^k - x^k))).
G.f.: exp(Sum_{k>=1} A000593(k)*x^k/(k*(1 + x)^k)).
From Peter Bala, Dec 22 2020: (Start)
O.g.f.: Sum_{n >= 0}  x^(n*(n+1)/2)/Product_{k = 1..n} ((1 + x)^k - x^k). Cf. A307548.
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)
a(n+1) = Sum_{k=0..n} (-1)^(n-k) binomial(n,k) A000009(k+1). - Gus Wiseman, Feb 03 2025

A320586 Expansion of (1/(1 - x)) * Sum_{k>=1} k*x^k/(x^k + (1 - x)^k).

Original entry on oeis.org

1, 3, 10, 27, 66, 156, 365, 843, 1909, 4238, 9274, 20136, 43564, 94013, 202155, 432475, 919820, 1945767, 4098852, 8610922, 18061277, 37844128, 79212323, 165565920, 345412341, 719047566, 1493488927, 3095654281, 6405734456, 13238611241, 27336762272, 56416256443, 116376652600

Offset: 1

Ilya Gutkovskiy, Oct 16 2018

nonn

Binomial transform of A000593.

G. C. Greubel, Table of n, a(n) for n = 1..1000
Vaclav Kotesovec, Plot of a(n)/(n*2^n) for n = 1..10000
N. J. A. Sloane, Transforms

Cf. A000593, A129519, A185003, A320589.

m:=50; R:=PowerSeriesRing(Rationals(), m); Coefficients(R!( (1/(1 +-x))*(&+[k*x^k/(x^k + (1 - x)^k): k in [1..(m+2)]]) )); // G. C. Greubel, Oct 30 2018

seq(coeff(series((1/(1-x))*add(k*x^k/(x^k+(1-x)^k),k=1..n),x,n+1), x, n), n = 1 .. 35); # Muniru A Asiru, Oct 16 2018

nmax = 33; Rest[CoefficientList[Series[1/(1 - x) Sum[k x^k/(x^k + (1 - x)^k), {k, 1, nmax}], {x, 0, nmax}], x]]
nmax = 33; Rest[CoefficientList[Series[(EllipticTheta[3, 0, x/(1 - x)]^4 + EllipticTheta[2, 0, x/(1 - x)]^4 - 1)/(24 (1 - x)), {x, 0, nmax}], x]]
Table[Sum[Binomial[n, k] Sum[(-1)^(k/d + 1) d, {d, Divisors[k]}], {k, n}], {n, 33}]

m=50; x='x+O('x^m); Vec((1/(1 - x))*sum(k=1, m+2, k*x^k/(x^k + (1 - x)^k))) \\ G. C. Greubel, Oct 30 2018

G.f.: (theta_3(x/(1 - x))^4 + theta_2(x/(1 - x))^4 - 1)/(24*(1 - x)), where theta_() is the Jacobi theta function.
L.g.f.: Sum_{k>=1} A000593(k)*x^k/(k*(1 - x)^k) = Sum_{n>=1} a(n)*x^n/n.
a(n) = Sum_{k=1..n} binomial(n,k)*A000593(k).
Conjecture: a(n) ~ c * 2^n * n, where c = Pi^2/24 = 0.411233516712... - Vaclav Kotesovec, Jun 26 2019

cp's OEIS Frontend

A320591 Expansion of Product_{k>=1} (1 + x^k/(1 + x)^k).

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