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

%I A318179 #17 Feb 16 2025 08:33:56
%S A318179 1,1,-3,5,25,-343,2133,-3603,-112975,1938897,-18008275,55198805,
%T A318179 1753746377,-45801271943,649021707397,-4682002329795,-50792700319903,
%U A318179 2692784088681889,-59182401177647011,801759226622986917,-2169423359710146183,-263145142263538606519,9869607872225170545333
%N A318179 Expansion of e.g.f. exp((1 - exp(-4*x))/4).
%H A318179 Seiichi Manyama, <a href="/A318179/b318179.txt">Table of n, a(n) for n = 0..474</a>
%H A318179 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/BellPolynomial.html">Bell Polynomial</a>
%F A318179 a(n) = Sum_{k=0..n} (-4)^(n-k)*Stirling2(n,k).
%F A318179 a(0) = 1; a(n) = Sum_{k=1..n} (-4)^(k-1)*binomial(n-1,k-1)*a(n-k).
%F A318179 a(n) = (-4)^n*BellPolynomial_n(-1/4). - _Peter Luschny_, Aug 20 2018
%p A318179 seq(n!*coeff(series(exp((1-exp(-4*x))/4),x=0,23),x,n),n=0..22); # _Paolo P. Lava_, Jan 09 2019
%t A318179 nmax = 22; CoefficientList[Series[Exp[(1 - Exp[-4 x])/4], {x, 0, nmax}], x] Range[0, nmax]!
%t A318179 Table[Sum[(-4)^(n - k) StirlingS2[n, k], {k, 0, n}], {n, 0, 22}]
%t A318179 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}]
%t A318179 Table[(-4)^n BellB[n, -1/4], {n, 0, 22}] (* _Peter Luschny_, Aug 20 2018 *)
%Y A318179 Column k=4 of A309386.
%Y A318179 Cf. A004213, A007696, A009235, A014182, A317996, A318180, A318181.
%K A318179 sign
%O A318179 0,3
%A A318179 _Ilya Gutkovskiy_, Aug 20 2018