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A201365 Expansion of e.g.f. exp(x) / (5 - 4*exp(x)).

%I A201365 #58 May 13 2025 20:42:14
%S A201365 1,5,45,605,10845,243005,6534045,204972605,7348546845,296387331005,
%T A201365 13282361478045,654762261324605,35211177242722845,2051349014835939005,
%U A201365 128701394409842982045,8651475271312083756605,620334325261670875138845,47259638324026516284867005
%N A201365 Expansion of e.g.f. exp(x) / (5 - 4*exp(x)).
%H A201365 G. C. Greubel, <a href="/A201365/b201365.txt">Table of n, a(n) for n = 0..358</a>
%F A201365 O.g.f.: A(x) = Sum_{n>=0} n! * 5^n*x^n / Product_{k=0..n} (1+k*x).
%F A201365 O.g.f.: A(x) = 1/(1 - 5*x/(1-4*x/(1 - 10*x/(1-8*x/(1 - 15*x/(1-12*x/(1 - 20*x/(1-16*x/(1 - 25*x/(1-20*x/(1 - ...))))))))))), a continued fraction.
%F A201365 a(n) = Sum_{k=0..n} (-1)^(n-k) * 5^k * Stirling2(n,k) * k!.
%F A201365 a(n) = Sum_{k=0..n} A123125(n,k)*5^k*4^(n-k). - _Philippe Deléham_, Nov 30 2011
%F A201365 a(n) ~ n! / (4*(log(5/4))^(n+1)) . - _Vaclav Kotesovec_, Jun 13 2013
%F A201365 a(n) = log(5/4) * Integral_{x = 0..oo} (ceiling(x))^n * (5/4)^(-x) dx. - _Peter Bala_, Feb 14 2015
%F A201365 a(n) = (1/4) Sum_{k>=1} (4/5)^k * n^k. - _Michael Somos_, Apr 27 2019
%F A201365 a(n) = 1 + 4 * Sum_{k=0..n-1} binomial(n,k) * a(k). - _Ilya Gutkovskiy_, Jun 08 2020
%F A201365 From _Seiichi Manyama_, Nov 15 2023: (Start)
%F A201365 a(0) = 1; a(n) = -5*Sum_{k=1..n} (-1)^k * binomial(n,k) * a(n-k).
%F A201365 a(0) = 1; a(n) = 5*a(n-1) + 4*Sum_{k=1..n-1} binomial(n-1,k) * a(n-k). (End)
%F A201365 a(n) = (5/4)*A094417(n) - (1/4)*0^n. - _Seiichi Manyama_, Dec 21 2023
%e A201365 E.g.f.: E(x) = 1 + 5*x + 45*x^2/2! + 605*x^3/3! + 10845*x^4/4! + 243005*x^5/5! + ...
%e A201365 O.g.f.: A(x) = 1 + 5*x + 45*x^2 + 605*x^3 + 10845*x^4 + 243005*x^5 + ...
%e A201365 where A(x) = 1 + 5*x/(1+x) + 2!*5^2*x^2/((1+x)*(1+2*x)) + 3!*5^3*x^3/((1+x)*(1+2*x)*(1+3*x)) + 4!*5^4*x^4/((1+x)*(1+2*x)*(1+3*x)*(1+4*x)) + ...
%p A201365 seq(coeff(series(1/(5*exp(-x) - 4), x, n+1)*n!, x, n), n = 0..20); # _G. C. Greubel_, Jun 08 2020
%t A201365 Table[Sum[(-1)^(n-k)*5^k*StirlingS2[n,k]*k!,{k,0,n}],{n,0,20}] (* _Vaclav Kotesovec_, Jun 13 2013 *)
%t A201365 With[{nn=20},CoefficientList[Series[Exp[x]/(5-4Exp[x]),{x,0,nn}],x] Range[ 0,nn]!] (* _Harvey P. Dale_, Jul 09 2015 *)
%t A201365 a[n_]:= If[n<0, 0, PolyLog[ -n, 4/5]/4]; (* _Michael Somos_, Apr 27 2019 *)
%o A201365 (PARI) {a(n)=n!*polcoeff(exp(x+x*O(x^n))/(5 - 4*exp(x+x*O(x^n))), n)}
%o A201365 (PARI) {a(n)=polcoeff(sum(m=0, n, 5^m*m!*x^m/prod(k=1, m, 1+k*x+x*O(x^n))), n)}
%o A201365 (PARI) {a(n)=sum(k=0, n, (-1)^(n-k)*5^k*stirling(n, k, 2)*k!)}
%o A201365 (Magma) R<x>:=PowerSeriesRing(Rationals(), 20); Coefficients(R!(Laplace( 1/(5*Exp(-x) -4) ))); // _G. C. Greubel_, Jun 08 2020
%o A201365 (Sage) [sum( (-1)^(n-j)*5^j*factorial(j)*stirling_number2(n,j) for j in (0..n)) for n in (0..20)] # _G. C. Greubel_, Jun 08 2020
%Y A201365 Cf. A027882, A032183, A050353.
%Y A201365 Cf. A201366, A201367, A201368.
%Y A201365 Cf. A000629, A201339, A201354.
%K A201365 nonn,easy
%O A201365 0,2
%A A201365 _Paul D. Hanna_, Nov 30 2011