A201339 - OEIS

%I #46 Dec 21 2023 10:21:55

%S 1,3,15,111,1095,13503,199815,3449631,68062695,1510769343,37260156615,

%T 1010843385951,29916558512295,959183053936383,33118910817665415,

%U 1225219266296167071,48348200298184769895,2027102674516399522623,89990106205541777926215,4216915299772659459872991

%N Expansion of e.g.f. exp(x) / (3 - 2*exp(x)).

%H G. C. Greubel, <a href="/A201339/b201339.txt">Table of n, a(n) for n = 0..385</a>

%F O.g.f.: A(x) = Sum_{n>=0} n! * 3^n*x^n / Product_{k=0..n} (1+k*x).

%F O.g.f.: A(x) = 1/(1 - 3*x/(1-2*x/(1 - 6*x/(1-4*x/(1 - 9*x/(1-6*x/(1 - 12*x/(1-8*x/(1 - 15*x/(1-10*x/(1 - ...)))))))))), a continued fraction.

%F a(n) = Sum_{k=0..n} (-1)^(n-k) * 3^k * Stirling2(n,k) * k!.

%F a(n) = 3*A050351(n) for n>0.

%F a(n) = Sum_{k=0..n} A123125(n,k)*3^k*2^(n-k). - _Philippe Deléham_, Nov 30 2011

%F a(n) ~ n! / (2*log(3/2)^(n+1)). - _Vaclav Kotesovec_, Jun 13 2013

%F a(n) = log(3/2) * Integral_{x = 0..oo} (ceiling(x))^n * (3/2)^(-x) dx. - _Peter Bala_, Feb 06 2015

%F a(n) = 1 + 2 * Sum_{k=0..n-1} binomial(n,k) * a(k). - _Ilya Gutkovskiy_, Jun 08 2020

%F From _Seiichi Manyama_, Nov 15 2023: (Start)

%F a(0) = 1; a(n) = -3*Sum_{k=1..n} (-1)^k * binomial(n,k) * a(n-k).

%F a(0) = 1; a(n) = 3*a(n-1) + 2*Sum_{k=1..n-1} binomial(n-1,k) * a(n-k). (End)

%F a(n) = (3/2)*A004123(n+1) - (1/2)*0^n. - _Seiichi Manyama_, Dec 21 2023

%e E.g.f.: E(x) = 1 + 3*x + 15*x^2/2! + 111*x^3/3! + 1095*x^4/4! + 13503*x^5/5! + ...

%e O.g.f.: A(x) = 1 + 3*x + 15*x^2 + 111*x^3 + 1095*x^4 + 13503*x^5 + ...

%e where A(x) = 1 + 3*x/(1+x) + 2!*3^2*x^2/((1+x)*(1+2*x)) + 3!*3^3*x^3/((1+x)*(1+2*x)*(1+3*x)) + 4!*3^4*x^4/((1+x)*(1+2*x)*(1+3*x)*(1+4*x)) + ...

%p seq(coeff(series( 1/(3*exp(-x) -2) , x, n+1)*n!, x, n), n = 0..30); # _G. C. Greubel_, Jun 08 2020

%t Table[Sum[(-1)^(n-k)*3^k*StirlingS2[n,k]*k!,{k,0,n}],{n,0,20}] (* _Vaclav Kotesovec_, Jun 13 2013 *)

%o (PARI) {a(n)=n!*polcoeff(exp(x+x*O(x^n))/(3 - 2*exp(x+x*O(x^n))), n)}

%o (PARI) {a(n)=polcoeff(sum(m=0, n, 3^m*m!*x^m/prod(k=1, m, 1+k*x+x*O(x^n))), n)}

%o (PARI) {Stirling2(n, k)=if(k<0|k>n, 0, sum(i=0, k, (-1)^i*binomial(k, i)/k!*(k-i)^n))}

%o {a(n)=sum(k=0, n, (-1)^(n-k)*3^k*Stirling2(n, k)*k!)}

%o (Magma) [&+[(-1)^(n-j)*3^j*Factorial(j)*StirlingSecond(n,j): j in [0..n]]: n in [0..20]]; // _G. C. Greubel_, Jun 08 2020

%o (Sage) [sum( (-1)^(n-j)*3^j*factorial(j)*stirling_number2(n,j) for j in (0..n)) for n in (0..20)] # _G. C. Greubel_, Jun 08 2020

%Y Cf. A000629, A004123, A050351, A123227.

%K nonn,easy

%O 0,2

%A _Paul D. Hanna_, Nov 30 2011