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

1, 1, 1, 0, 1, -2, 5, -12, 28, -63, 137, -290, 604, -1253, 2617, -5537, 11870, -25666, 55617, -120103, 257582, -548119, 1158437, -2437114, 5117165, -10748530, 22621055, -47728657, 100932549, -213750621, 452855190, -958925784, 2028187595, -4283531490, 9033779224

Offset: 0

Author

Ilya Gutkovskiy, Oct 16 2018

Keywords

• sign

Comments

The zero-based binomial transform of this sequence is A000070, and if we remove first terms it becomes A000041.

Links

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

Crossrefs

Cf. A000203, A103446, A218482, A320568.

Row n=1 of A175804 (except first term). Row n=0 is A281425.

The version for strict partitions is A320591, row n=1 of A378622, first column A293467.

A000009 counts strict integer partitions, differences A087897, A378972.

A000041 counts integer partitions, differences A002865.

Cf. A047966, A008284, A377056, A378621.

Programs

Magma

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

Maple

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

Mathematica

nmax = 34; CoefficientList[Series[Product[1/(1 - x^k/(1 + x)^k), {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 34; CoefficientList[Series[Exp[Sum[DivisorSigma[1, k] x^k/(k (1 + x)^k), {k, 1, nmax}]], {x, 0, nmax}], x]

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

Formula

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

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