A354237 Expansion of e.g.f. 1 / (1 - log(1 + 2*x) / 2).

1, 1, 0, 2, -8, 64, -592, 6768, -90624, 1395840, -24292608, 471453696, -10094066688, 236340378624, -6007053852672, 164713554069504, -4846361933021184, 152300800682754048, -5091189648734748672, 180386551596145508352, -6752521487083688165376

Offset

0,4

Links

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

Formula

a(n) = Sum_{k=0..n} Stirling1(n,k) * k! * 2^(n-k).

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

a(n) ~ n! * (-1)^(n+1) * 2^(n+1) / (n * log(n)^2) * (1 - (4 + 2*gamma)/log(n) + (12 + 12*gamma + 3*gamma^2 - Pi^2/2)/log(n)^2 + (2*Pi^2*gamma - 32 + 4*Pi^2 - 24*gamma^2 - 8*zeta(3) - 4*gamma^3 - 48*gamma)/log(n)^3 + (80 - 20*Pi^2*gamma + 40*zeta(3)*gamma - 5*Pi^2*gamma^2 + 160*gamma + 5*gamma^4 + 80*zeta(3) + 40*gamma^3 + Pi^4/12 - 20*Pi^2 + 120*gamma^2)/log(n)^4), where gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, Jun 06 2022

Mathematica

nmax = 20; CoefficientList[Series[1/(1 - Log[1 + 2 x]/2), {x, 0, nmax}], x] Range[0, nmax]!
Table[Sum[StirlingS1[n, k] k! 2^(n - k), {k, 0, n}], {n, 0, 20}]

Prog

(PARI) my(x='x + O('x^20)); Vec(serlaplace(1/(1-log(1+2*x)/2))) \\ Michel Marcus, Jun 06 2022

Crossrefs

Cf. A006252, A088500, A088501, A122704, A155585, A227917, A354750, A354751.

Sequence in context: A323853 A354276 A353821 * A191570 A052707 A059862

Adjacent sequences: A354234 A354235 A354236 * A354238 A354239 A354240

Keyword

sign

Author

Ilya Gutkovskiy, Jun 06 2022

Status

approved