A318179 Expansion of e.g.f. exp((1 - exp(-4*x))/4).

1, 1, -3, 5, 25, -343, 2133, -3603, -112975, 1938897, -18008275, 55198805, 1753746377, -45801271943, 649021707397, -4682002329795, -50792700319903, 2692784088681889, -59182401177647011, 801759226622986917, -2169423359710146183, -263145142263538606519, 9869607872225170545333

Offset: 0

Author

Ilya Gutkovskiy, Aug 20 2018

Keywords

• sign

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..474
• Eric Weisstein's World of Mathematics, Bell Polynomial

Crossrefs

Column k=4 of A309386.

Cf. A004213, A007696, A009235, A014182, A317996, A318180, A318181.

Programs

Maple

seq(n!*coeff(series(exp((1-exp(-4*x))/4),x=0,23),x,n),n=0..22); # Paolo P. Lava, Jan 09 2019

Mathematica

nmax = 22; CoefficientList[Series[Exp[(1 - Exp[-4 x])/4], {x, 0, nmax}], x] Range[0, nmax]!
Table[Sum[(-4)^(n - k) StirlingS2[n, k], {k, 0, n}], {n, 0, 22}]
a[n_] := a[n] = Sum[(-4)^(k - 1) Binomial[n - 1, k - 1] a[n - k], {k, 1, n}]; a[0] = 1; Table[a[n], {n, 0, 22}]
Table[(-4)^n BellB[n, -1/4], {n, 0, 22}] (* Peter Luschny, Aug 20 2018 *)

Formula

a(n) = Sum_{k=0..n} (-4)^(n-k)*Stirling2(n,k).

a(0) = 1; a(n) = Sum_{k=1..n} (-4)^(k-1)*binomial(n-1,k-1)*a(n-k).

a(n) = (-4)^n*BellPolynomial_n(-1/4). - Peter Luschny, Aug 20 2018