A234643 - OEIS

A234643

E.g.f.: Sum_{n>=0} Integral^n (exp(x) + 1)^n dx^n, where integral^n F(x) dx^n is the n-th integration of F(x) with no constant of integration.

%I #7 Dec 29 2013 04:21:35

%S 1,2,5,13,35,99,297,951,3265,12047,47761,202975,921281,4447327,

%T 22737537,122639583,695404929,4132531679,25667031937,166211936735,

%U 1119791799425,7833568488415,56802921911681,426267651506655,3305731721387649,26457699508131807,218276886237532033

%N E.g.f.: Sum_{n>=0} Integral^n (exp(x) + 1)^n dx^n, where integral^n F(x) dx^n is the n-th integration of F(x) with no constant of integration.

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

%e E.g.f.: A(x) = 1 + 2*x + 5*x^2/2! + 13*x^3/3! + 35*x^4/4! + 99*x^5/5! +...

%e where the e.g.f. may be expressed as a series involving iterated integration:

%e A(x) = 1 + Integral (exp(x)+1) dx + Integral^2 (exp(x)+1)^2 dx^2 + Integral^3 (exp(x)+1)^3 dx^3 + Integral^4 (exp(x)+1)^4 dx^4 +...

%o (PARI) {a(n)=sum(k=0,n, sum(j=0,k,binomial(k,j)*j^(n-k)))}

%o for(n=0,30,print1(a(n),", "))

%o (PARI) {INTEGRATE(n,F)=local(G=F);for(i=1,n,G=intformal(G));G}

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

%o for(n=0,30,print1(a(n),", "))

%Y Cf. A105795.

%K nonn

%O 0,2

%A _Paul D. Hanna_, Dec 29 2013