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%I #54 Jan 15 2024 00:51:44
%S 1,4,36,484,8676,194404,5227236,163978084,5878837476,237109864804,
%T 10625889182436,523809809059684,28168941794178276,1641079211868751204,
%U 102961115527874385636,6921180217049667005284,496267460209336700111076
%N Generalized ordered Bell numbers Bo(4,n).
%C Fourth row of array A094416, which has more information.
%H Vincenzo Librandi, <a href="/A094417/b094417.txt">Table of n, a(n) for n = 0..200</a>
%H Paul Barry, <a href="https://arxiv.org/abs/1803.06408">Three Études on a sequence transformation pipeline</a>, arXiv:1803.06408 [math.CO], 2018.
%F E.g.f.: 1/(5 - 4*exp(x)).
%F a(n) = 4 * A050353(n) for n>0.
%F a(n) = Sum_{k=0..n} A131689(n,k)*4^k. - _Philippe Deléham_, Nov 03 2008
%F E.g.f.: A(x) with A_n = 4 * Sum_{k=0..n-1} C(n,k) * A_k; A_0 = 1. - _Vladimir Kruchinin_, Jan 27 2011
%F G.f.: 2/G(0), where G(k)= 1 + 1/(1 - 8*x*(k+1)/(8*x*(k+1) - 1 + 10*x*(k+1)/G(k+1))); (continued fraction). - _Sergei N. Gladkovskii_, May 30 2013
%F a(n) = log(5/4)*int {x = 0..inf} (floor(x))^n * (5/4)^(-x) dx. - _Peter Bala_, Feb 14 2015
%F a(0) = 1; a(n) = 4*a(n-1) - 5*Sum_{k=1..n-1} (-1)^k * binomial(n-1,k) * a(n-k). - _Seiichi Manyama_, Nov 16 2023
%p a:= proc(n) option remember;
%p `if`(n=0, 1, 4* add(binomial(n, k) *a(k), k=0..n-1))
%p end:
%p seq(a(n), n=0..20);
%t max = 16; f[x_] := 1/(5-4*E^x); CoefficientList[Series[f[x], {x, 0, max}], x]*Range[0, max]! (* _Jean-François Alcover_, Nov 14 2011, after g.f. *)
%o (Magma) m:=20; R<x>:=LaurentSeriesRing(RationalField(), m); b:=Coefficients(R!(1/(5 - 4*Exp(x)))); [Factorial(n-1)*b[n]: n in [1..m]]; // _Bruno Berselli_, Mar 17 2014
%o (SageMath)
%o def A094416(n,k): return sum(factorial(j)*n^j*stirling_number2(k,j) for j in range(k+1)) # array
%o def A094417(k): return A094416(4,k)
%o [A094417(n) for n in range(31)] # _G. C. Greubel_, Jan 12 2024
%o (PARI) my(N=25,x='x+O('x^N)); Vec(serlaplace(1/(5 - 4*exp(x)))) \\ _Joerg Arndt_, Jan 15 2024
%Y Cf. A000670, A004123, A032033, A050353, A094418, A094419, A131689.
%Y Cf. A346983, A354242, A365567.
%K nonn
%O 0,2
%A _Ralf Stephan_, May 02 2004
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