A344559 a(n) = (1/6) * 2^(-n) * n! * [x^n] Exp(2*x, 1)*(Exp(2*x, 3) - 1), where Exp(x, m) = Sum_{k>=0} (x^k / k!)^m.

0, 0, 0, 1, 4, 10, 35, 140, 476, 1624, 6070, 22495, 81455, 301301, 1131494, 4230681, 15852396, 59881956, 226877648, 860447129, 3273728234, 12493453344, 47760610689, 182905145214, 701883651799, 2697952583635, 10385325566785, 40033903418860, 154534663044346

Offset: 0

Author

Peter Luschny, Jun 01 2021

Keywords

• nonn

Crossrefs

Cf. A344854.

Programs

Maple

Exp := (x, m) -> sum((x^k / k!)^m, k=0..infinity):
gf := Exp(2*x, 1)*(Exp(2*x, 3) - 1): ser := series(gf, x, 34):
seq((1/6)*2^(-n)*n!*simplify(coeff(ser, x, n)), n = 0..28);

Mathematica

a[n_] := (1/6) (HypergeometricPFQ[{-n/3, (1 - n)/3, (2 - n)/3}, {1, 1}, -27] - 1);
Table[a[n], {n, 0, 28}]

Python

from sympy import hyperexpand, Rational
from sympy.functions import hyper
def A344559(n): return (hyperexpand(hyper((Rational(-n,3),Rational(1-n,3),Rational(2-n,3)),(1,1),-27))-1)//6 # Chai Wah Wu, Jan 04 2024

Formula

a(n) = A344854(n) / 2^n.

a(n) = (1/6)*(hypergeom([-n/3, (1 - n)/3, (2 - n)/3], [1, 1], -27) - 1).