A293565 Expansion of e.g.f. Product_{k>=0} exp(-x^(3*k+1)).

1, -1, 1, -1, -23, 119, -359, -4201, 58801, -359857, -2121839, 74851919, -784376999, -1072414201, 192596540137, -3167640042841, 9241215940321, 860635956717599, -21290802661288799, 157132294452113183, 5980504635868174921, -220612220726623344361

Offset

0,5

Links

Seiichi Manyama, Table of n, a(n) for n = 0..447

Formula

E.g.f.: exp(x/(x^3 - 1)).

a(0) = 1; a(n) = -Sum_{k=0..floor((n-1)/3)} binomial(n-1,3*k) * (3*k+1)! * a(n-3*k-1). - Ilya Gutkovskiy, Feb 24 2022

a(n) = n! * Sum_{k=0..floor(n/3)} (-1)^(n-k) * binomial(n-2*k-1,k)/(n-3*k)!. - Seiichi Manyama, Jun 08 2024

Mathematica

CoefficientList[Series[E^(x/(x^3 - 1)), {x, 0, 20}], x] * Range[0, 20]! (* Vaclav Kotesovec, Oct 13 2017 *)

Prog

(PARI) N=66; x='x+O('x^N); Vec(serlaplace(exp(x/(x^3-1))))
(PARI) N=66; x='x+O('x^N); Vec(serlaplace(1/prod(m=0, N, exp(x^(3*m+1)))))

Crossrefs

E.g.f.: Product_{m>=0} exp(-x^(k*m+1)): A293116 (k=1), A293532 (k=2), this sequence (k=3), A293566 (k=4).

Cf. A293493.

Sequence in context: A362339 A358063 A367722 * A362569 A099068 A220408

Adjacent sequences: A293562 A293563 A293564 * A293566 A293567 A293568

Keyword

sign

Author

Seiichi Manyama, Oct 12 2017

Status

approved