cp's OEIS Frontend

A217911 O.g.f.: Sum_{n>=0} n^n * (3*n+1)^(n-1) * exp(-n*(3*n+1)*x) * x^n / n!.

%I A217911 #15 Feb 13 2017 17:53:41
%S A217911 1,1,10,262,11296,684172,53598952,5162269744,590636585728,
%T A217911 78321222303184,11815503098606560,1998732510370890208,
%U A217911 374763163567227915520,77151431783218955979520,17301697176590720940003328,4198491769695976346962419712,1096165878182404669364316147712
%N A217911 O.g.f.: Sum_{n>=0} n^n * (3*n+1)^(n-1) * exp(-n*(3*n+1)*x) * x^n / n!.
%C A217911 Compare the g.f. to the LambertW identity:
%C A217911 1 = Sum_{n>=0} (3*n+1)^(n-1) * exp(-(3*n+1)*x) * x^n/n!.
%H A217911 G. C. Greubel, <a href="/A217911/b217911.txt">Table of n, a(n) for n = 0..300</a>
%F A217911 a(n) = 1/n! * Sum_{k=0..n} (-1)^(n-k)*binomial(n,k) * k^n * (3*k+1)^(n-1).
%F A217911 a(n) = 1/n! * [x^n] Sum_{k>=0} k^k*(3*k+1)^(k-1)*x^k / (1 + k*(3*k+1)*x)^(k+1).
%F A217911 a(n) = [x^n] 1 + x*(1+x)^(n-1) / Product_{k=1..n} (1 - 3*k*x).
%F A217911 a(n) = [x^n] 1 + x*(1-x)^(n-1) / Product_{k=1..n} (1 - (3*k+1)*x).
%F A217911 a(n) ~ 2^(2*n-4/3) * 3^(n-1) * n^(n-3/2) / (sqrt(Pi*(1-c)) * exp(n) * (2-c)^(n-1) * c^(n+1/6)), where c = -LambertW(-2*exp(-2)) = 0.4063757399599599... . - _Vaclav Kotesovec_, May 22 2014
%e A217911 O.g.f.: A(x) = 1 + x + 10*x^2 + 262*x^3 + 11296*x^4 + 684172*x^5 +...
%e A217911 where
%e A217911 A(x) = 1 + 1^1*4^0*x*exp(-1*4*x) + 2^2*7^1*exp(-2*7*x)*x^2/2! + 3^3*10^2*exp(-3*10*x)*x^3/3! + 4^4*13^3*exp(-4*13*x)*x^4/4! + 5^5*16^4*exp(-5*16*x)*x^5/5! +...
%e A217911 simplifies to a power series in x with integer coefficients.
%t A217911 Flatten[{1,Table[Sum[Binomial[n-1,j]*3^j*StirlingS2[n+j,n],{j,0,n-1}],{n,1,20}]}] (* _Vaclav Kotesovec_, May 22 2014 *)
%o A217911 (PARI) {a(n)=polcoeff(sum(k=0,n,k^k*(3*k+1)^(k-1)*x^k*exp(-k*(3*k+1)*x+x*O(x^n))/k!),n)}
%o A217911 for(n=0,30,print1(a(n),", "))
%o A217911 (PARI) {a(n)=1/n!*polcoeff(sum(k=0, n, k^k*(3*k+1)^(k-1)*x^k/(1+k*(3*k+1)*x +x*O(x^n))^(k+1)), n)}
%o A217911 (PARI) {a(n)=1/n!*sum(k=0,n, (-1)^(n-k)*binomial(n,k)*k^n*(3*k+1)^(n-1))}
%o A217911 for(n=0,30,print1(a(n),", "))
%o A217911 (PARI) {a(n)=polcoeff(1+x*(1+x)^(n-1)/prod(k=0, n, 1-3*k*x +x*O(x^n)), n)}
%o A217911 for(n=0,30,print1(a(n),", "))
%o A217911 (PARI) {a(n)=polcoeff(1+x*(1-x)^n/prod(k=0, n, 1-(3*k+1)*x +x*O(x^n)), n)}
%o A217911 for(n=0,30,print1(a(n),", "))
%Y A217911 Cf. A217900, A217910, A217912.
%K A217911 nonn
%O A217911 0,3
%A A217911 _Paul D. Hanna_, Oct 14 2012