A248653 - OEIS

%I #6 Oct 18 2014 12:41:43

%S 1,3,20,237,4276,107225,3518526,145005721,7285611096,436297841649,

%T 30590014543930,2474931380486081,228308751882636756,

%U 23772216923031342649,2769853988736186166374,358463639909150646730665,51192480930691715108562736,8021370202848006225125239649

%N E.g.f.: Sum_{n>=0} x^n * (2 + exp(n*x))^n.

%F E.g.f.: Sum_{n>=0} x^n * exp(n^2*x) / (1 - 2*x*exp(n*x))^(n+1).

%e G.f.: A(x) = 1 + 3*x + 20*x^2/2! + 237*x^3/3! + 4276*x^4/4! + 107225*x^5/5! +...

%e where the g.f. satisfies following series identity:

%e A(x) = 1 + x*(2+exp(x)) + x^2*(2+exp(2*x))^2 + x^3*(2+exp(3*x))^3 + x^4*(2+exp(4*x))^4 + x^5*(2+exp(5*x))^5 + x^6*(2+exp(6*x))^6 +...

%e A(x) = 1/(1-2*x) + x*exp(x)/(1-2*x*exp(x))^2 + x^2*exp(4*x)/(1-2*x*exp(2*x))^3 + x^3*exp(9*x)/(1-2*x*exp(3*x))^4 + x^4*exp(16*x)/(1-2*x*exp(4*x))^5 + x^5*exp(25*x)/(1-2*x*exp(5*x))^6 + x^6*exp(36*x)/(1-2*x*exp(6*x))^7 +...

%o (PARI) {a(n,t=2)=local(A=1); A=sum(k=0, n, x^k * (t + exp(k*x +x*O(x^n)))^k); n!*polcoeff(A, n)}

%o for(n=0,25,print1(a(n,2),", "))

%o (PARI) {a(n,t=2)=local(A=1); A=sum(k=0, n, x^k * exp(k^2*x +x*O(x^n))/(1 - t*x*exp(k*x +x*O(x^n)))^(k+1) ); n!*polcoeff(A, n)}

%o for(n=0,25,print1(a(n,2),", "))

%Y Cf. A248615, A193421, A248654.

%K nonn

%O 0,2

%A _Paul D. Hanna_, Oct 18 2014