A355372 Expansion of the e.g.f. log((1 - x) / (1 - 2*x)) / (1 - x)^3.

0, 1, 9, 77, 714, 7374, 85272, 1102968, 15908400, 254866320, 4516084800, 88102382400, 1883199024000, 43885950595200, 1109416142822400, 30273281955302400, 887493144729139200, 27827941161784780800, 929449073791558656000, 32943696020637889536000, 1234946945823695419392000

Offset: 0

Author

Mélika Tebni, Jun 30 2022

Keywords

• nonn

Comments

Conjecture: For p prime, a(p) == -1 (mod p).

Crossrefs

Cf. A000292, A062139, A355171.

Programs

Maple

A355372 := n -> A000292(n)*n!*hypergeom([1 - n, 1, 1], [2, 4], -1):
seq(simplify(A355372(n)), n = 0..20);

Mathematica

CoefficientList[Series[Log[(1 - x)/(1 - 2*x)]/ (1 - x)^3,{x,0,20}],x]Table[n!,{n,0,20}] (* Stefano Spezia, Jun 30 2022 *)

Formula

a(n) = Sum_{k=0..n} (-1)^(k+1)*k!*A062139(n, k + 1).

a(0) = 0, a(n) = n!*Sum_{k=1..n} (n-k+2)*(n-k+1)*(2^k-1)/(2*k).

a(n) = A000292(n)*n!*hypergeom([1 - n, 1, 1], [2, 4], -1). - Peter Luschny, Jun 30 2022