A355407 Expansion of the e.g.f. log((1 - x) / (1 - 2*x)) / (1 - x)^4.

0, 1, 11, 110, 1154, 13144, 164136, 2251920, 33923760, 560180160, 10117886400, 199399132800, 4275988617600, 99473802624000, 2502049379558400, 67804022648678400, 1972357507107993600, 61358018782620672000, 2033893411878730752000, 71587670846333773824000, 2666700362750370895872000

Offset: 0

Author

Mélika Tebni, Jul 01 2022

Keywords

• nonn

Comments

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

Crossrefs

Cf. A000292, A000332, A062137, A355171, A355372.

Programs

Maple

egf := log((1 - x)/(1 - 2*x))/(1 - x)^4: ser := series(egf, x, 22):
seq(n!*coeff(ser, x, n), n = 0..20); # Peter Luschny, Jul 01 2022

Mathematica

With[{nn=20},CoefficientList[Series[Log[((1-x)/(1-2x))]/(1-x)^4,{x,0,nn}],x] Range[ 0,nn]!] (* Harvey P. Dale, Mar 09 2023 *)

Formula

a(n) = Sum_{k=0..n} (-1)^(k+1)*k!*A062137(n, k+1).

a(0) = 0, a(n) = n!*Sum_{k=1..n} A000292(n-k+1)*(2^k-1)/k.

a(n) = A000332(n+3)*n!*hypergeom([1 - n, 1, 1], [2, 5], -1). - Peter Luschny, Jul 01 2022