A307500 Expansion of Product_{k>=1} 1/(1 - (x*(1 - x))^k).

1, 1, 1, -1, -2, -4, 3, -1, 17, -16, 21, -57, 67, -130, 305, -536, 995, -1726, 2652, -4286, 7320, -13043, 24458, -45405, 81415, -141724, 239755, -400603, 676872, -1171076, 2072334, -3695550, 6519951, -11279015, 19188230, -32462795, 55334284, -95718737, 167673672, -294894076

Offset

0,5

Links

Seiichi Manyama, Table of n, a(n) for n = 0..1000

Formula

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

a(n) = Sum_{k=0..n} (-1)^(n-k)*binomial(k,n-k)*A000041(k).

Mathematica

nmax = 39; CoefficientList[Series[Product[1/(1 - (x (1 - x))^k), {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 39; CoefficientList[Series[Exp[Sum[DivisorSigma[1, k] (x (1 - x))^k/k, {k, 1, nmax}]], {x, 0, nmax}], x]
Table[Sum[(-1)^(n - k) Binomial[k, n - k] PartitionsP[k], {k, 0, n}], {n, 0, 39}]

Crossrefs

Cf. A000041, A030528, A218482, A238441, A286955, A307501.

Sequence in context: A322398 A341606 A109158 * A049245 A123547 A123551

Adjacent sequences: A307497 A307498 A307499 * A307501 A307502 A307503

Keyword

sign

Author

Ilya Gutkovskiy, Apr 11 2019

Status

approved