cp's OEIS Frontend

A217912 O.g.f.: Sum_{n>=0} 2*n^n * (3*n+2)^(n-1) * exp(-n*(3*n+2)*x) * x^n / n!.

%I A217912 #30 Apr 02 2017 03:04:00
%S A217912 1,2,22,602,26656,1643054,130318966,12666846728,1459524093232,
%T A217912 194626267782398,29495119281572770,5008297010070635978,
%U A217912 942044179147597185544,194462342099815302424136,43711609296992502659474632,10628894996508864880841838416,2780041837527932453797746700384
%N A217912 O.g.f.: Sum_{n>=0} 2*n^n * (3*n+2)^(n-1) * exp(-n*(3*n+2)*x) * x^n / n!.
%C A217912 Compare the g.f. to the LambertW identity:
%C A217912 1 = Sum_{n>=0} 2*(3*n+2)^(n-1) * exp(-(3*n+2)*x) * x^n/n!.
%H A217912 G. C. Greubel, <a href="/A217912/b217912.txt">Table of n, a(n) for n = 0..300</a>
%F A217912 a(n) = 1/n! * Sum_{k=0..n} 2*(-1)^(n-k)*binomial(n,k) * k^n * (3*k+2)^(n-1).
%F A217912 a(n) = 1/n! * [x^n] Sum_{k>=0} 2*k^k*(3*k+2)^(k-1)*x^k / (1 + k*(3*k+2)*x)^(k+1).
%F A217912 a(n) = [x^n] 1 + x*(1+2*x)^(n-1) / Product_{k=1..n} (1 - 3*k*x).
%F A217912 a(n) = [x^n] 1 + x*(1-2*x)^(n-1) / Product_{k=1..n} (1 - (3*k+2)*x).
%F A217912 a(n) ~ 2^(2*n-1/6) * 3^(n-1) * n^(n-3/2) / (sqrt(Pi*(1-c)) * exp(n) * (2-c)^(n-1) * c^(n+1/3)), where c = -LambertW(-2*exp(-2)) = 0.4063757399599599... . - _Vaclav Kotesovec_, May 22 2014
%e A217912 O.g.f.: A(x) = 1 + 2*x + 22*x^2 + 602*x^3 + 26656*x^4 + 1643054*x^5 + ...
%e A217912 where
%e A217912 A(x) = 1 + 2*1^1*5^0*x*exp(-1*5*x) + 2*2^2*8^1*exp(-2*8*x)*x^2/2! + 2*3^3*11^2*exp(-3*11*x)*x^3/3! + 2*4^4*14^3*exp(-4*14*x)*x^4/4! + 2*5^5*17^4*exp(-5*17*x)*x^5/5! + ...
%e A217912 simplifies to a power series in x with integer coefficients.
%t A217912 Flatten[{1,Table[Sum[Binomial[n-1,j]*3^j*2^(n-j)*StirlingS2[n+j,n],{j,0,n-1}],{n,1,20}]}] (* _Vaclav Kotesovec_, May 22 2014 *)
%o A217912 (PARI) {a(n)=polcoeff(sum(k=0,n,2*k^k*(3*k+2)^(k-1)*x^k*exp(-k*(3*k+2)*x+x*O(x^n))/k!),n)}
%o A217912 for(n=0,30,print1(a(n),", "))
%o A217912 (PARI) {a(n)=1/n!*polcoeff(sum(k=0, n, 2*k^k*(3*k+2)^(k-1)*x^k/(1+k*(3*k+2)*x +x*O(x^n))^(k+1)), n)}
%o A217912 (PARI) {a(n)=1/n!*sum(k=0,n, 2*(-1)^(n-k)*binomial(n,k)*k^n*(3*k+2)^(n-1))}
%o A217912 for(n=0,30,print1(a(n),", "))
%o A217912 (PARI) {a(n)=polcoeff(1+2*x*(1+2*x)^(n-1)/prod(k=0, n, 1-3*k*x +x*O(x^n)), n)}
%o A217912 for(n=0,30,print1(a(n),", "))
%o A217912 (PARI) {a(n)=polcoeff(1+2*x*(1-2*x)^n/prod(k=0, n, 1-(3*k+2)*x +x*O(x^n)), n)}
%o A217912 for(n=0,30,print1(a(n),", "))
%o A217912 (Maxima) makelist( if n=0 then 1 else 1/n! * sum(2*(-1)^(n-k)*binomial(n,k) * k^n * (3*k+2)^(n-1),k,0,n), n, 0, 30); /* _Martin Ettl_, Oct 15 2012 */
%Y A217912 Cf. A217900, A217910, A217911, A217913.
%K A217912 nonn
%O A217912 0,2
%A A217912 _Paul D. Hanna_, Oct 14 2012