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

0,3

Comments

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

The conjecture is true (see Fried link). - Sela Fried, Dec 17 2025

Links

Table of n, a(n) for n=0..20.

Sela Fried, Proof of a conjecture stated in A355372, 2025.

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

a(n) = n!*Sum_{k=1..n} binomial(n+2, k+2)/k. - Sela Fried, Dec 17 2025

a(n) = 3*n*a(n-1) - 2*(n^2-1)*a(n-2) + (n+1)!/2 for n > 1. - Mélika Tebni, Mar 20 2026

Maple

A355372 := n -> A000292(n)*n!*hypergeom([1 - n, 1, 1], [2, 4], -1):
seq(simplify(A355372(n)), n = 0..20);
# Alternative:
a := proc (n) option remember; `if`(n < 2, n, 3*n*a(n-1)-2*(n^2-1)*a(n-2)+(n+1)!/2) end proc:
seq(a(n), n = 0 .. 20); # Mélika Tebni, Mar 20 2026

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 *)

Crossrefs

Cf. A000292, A062139, A355171.

Sequence in context: A254659 A390682 A346847 * A046150 A124131 A000445

Adjacent sequences: A355369 A355370 A355371 * A355373 A355374 A355375

Keyword

nonn

Author

Mélika Tebni, Jun 30 2022

Status

approved