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A201354 Expansion of e.g.f. exp(x) / (4 - 3*exp(x)).

%I A201354 #55 May 28 2025 14:51:38
%S A201354 1,4,28,292,4060,70564,1471708,35810212,995827420,31153998244,
%T A201354 1082931514588,41407678132132,1727226633730780,78051253062575524,
%U A201354 3798351192214837468,198049421007186054052,11014905131587945490140,650903915009792820650404,40726453234725158535472348
%N A201354 Expansion of e.g.f. exp(x) / (4 - 3*exp(x)).
%H A201354 G. C. Greubel, <a href="/A201354/b201354.txt">Table of n, a(n) for n = 0..370</a>
%F A201354 O.g.f.: A(x) = Sum_{n>=0} n! * 4^n*x^n / Product_{k=0..n} (1+k*x).
%F A201354 O.g.f.: A(x) = 1/(1 - 4*x/(1-3*x/(1 - 8*x/(1-6*x/(1 - 12*x/(1-9*x/(1 - 16*x/(1-12*x/(1 - 20*x/(1-15*x/(1 - ...))))))))))), a continued fraction.
%F A201354 a(n) = Sum_{k=0..n} (-1)^(n-k) * 4^k * Stirling2(n,k) * k!.
%F A201354 a(n) = 4*A050352(n) for n>0.
%F A201354 a(n) = Sum_{k=0..n} A123125(n,k)*4^k*3^(n-k). - _Philippe Deléham_, Nov 30 2011
%F A201354 a(n) = log(4/3) * Integral_{x = 0..oo} (ceiling(x))^n * (4/3)^(-x) dx. - _Peter Bala_, Feb 06 2015
%F A201354 G.f.: 2/G(0), where G(k) = 1 + 1/(1 - 8*x*(k+1)/(8*x*(k+1) - 1 + 6*x*(k+1)/G(k+1))); (continued fraction). - _Sergei N. Gladkovskii_, May 30 2013
%F A201354 a(n) ~ n! / (3*(log(4/3))^(n+1)). - _Vaclav Kotesovec_, Jun 13 2013
%F A201354 a(n) = 1 + 3 * Sum_{k=0..n-1} binomial(n,k) * a(k). - _Ilya Gutkovskiy_, Jun 08 2020
%F A201354 From _Seiichi Manyama_, Nov 15 2023: (Start)
%F A201354 a(0) = 1; a(n) = -4*Sum_{k=1..n} (-1)^k * binomial(n,k) * a(n-k).
%F A201354 a(0) = 1; a(n) = 4*a(n-1) + 3*Sum_{k=1..n-1} binomial(n-1,k) * a(n-k). (End)
%F A201354 a(n) = (4/3)*A032033(n) - (1/3)*0^n. - _Seiichi Manyama_, Dec 21 2023
%e A201354 E.g.f.: E(x) = 1 + 4*x + 28*x^2/2! + 292*x^3/3! + 4060*x^4/4! + 70564*x^5/5! + ...
%e A201354 O.g.f.: A(x) = 1 + 4*x + 28*x^2 + 292*x^3 + 4060*x^4 + 70564*x^5 + ...
%e A201354 where A(x) = 1 + 4*x/(1+x) + 2!*4^2*x^2/((1+x)*(1+2*x)) + 3!*4^3*x^3/((1+x)*(1+2*x)*(1+3*x)) + 4!*4^4*x^4/((1+x)*(1+2*x)*(1+3*x)*(1+4*x)) + ...
%p A201354 seq(coeff(series(1/(4*exp(-x) -3), x, n+1)*n!, x, n), n = 0..20); # _G. C. Greubel_, Jun 08 2020
%t A201354 Table[Sum[(-1)^(n-k)*4^k*StirlingS2[n,k]*k!,{k,0,n}],{n,0,20}] (* _Vaclav Kotesovec_, Jun 13 2013 *)
%o A201354 (PARI) {a(n)=n!*polcoeff(exp(x+x*O(x^n))/(4 - 3*exp(x+x*O(x^n))), n)}
%o A201354 (PARI) {a(n)=polcoeff(sum(m=0, n, 4^m*m!*x^m/prod(k=1, m, 1+k*x+x*O(x^n))), n)}
%o A201354 (PARI) {Stirling2(n, k)=if(k<0||k>n, 0, sum(i=0, k, (-1)^i*binomial(k, i)/k!*(k-i)^n))}
%o A201354 {a(n)=sum(k=0, n, (-1)^(n-k)*4^k*Stirling2(n, k)*k!)}
%o A201354 (Magma) R<x>:=PowerSeriesRing(Rationals(), 20); Coefficients(R!(Laplace( 1/(4*Exp(-x) -3) ))); // _G. C. Greubel_, Jun 08 2020
%o A201354 (Sage) [sum( (-1)^(n-j)*4^j*factorial(j)*stirling_number2(n,j) for j in (0..n)) for n in (0..20)] # _G. C. Greubel_, Jun 08 2020
%Y A201354 Cf. A000629, A032033, A050352, A123227, A136728.
%K A201354 nonn,easy
%O A201354 0,2
%A A201354 _Paul D. Hanna_, Nov 30 2011