cp's OEIS Frontend

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 A234643 #7 Dec 29 2013 04:21:35
%S A234643 1,2,5,13,35,99,297,951,3265,12047,47761,202975,921281,4447327,
%T A234643 22737537,122639583,695404929,4132531679,25667031937,166211936735,
%U A234643 1119791799425,7833568488415,56802921911681,426267651506655,3305731721387649,26457699508131807,218276886237532033
%N 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.
%F A234643 a(n) = Sum_{k=0..n} Sum_{j=0..k} binomial(k,j) * j^(n-k).
%e A234643 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 A234643 where the e.g.f. may be expressed as a series involving iterated integration:
%e A234643 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 A234643 (PARI) {a(n)=sum(k=0,n, sum(j=0,k,binomial(k,j)*j^(n-k)))}
%o A234643 for(n=0,30,print1(a(n),", "))
%o A234643 (PARI) {INTEGRATE(n,F)=local(G=F);for(i=1,n,G=intformal(G));G}
%o A234643 {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 A234643 for(n=0,30,print1(a(n),", "))
%Y A234643 Cf. A105795.
%K A234643 nonn
%O A234643 0,2
%A A234643 _Paul D. Hanna_, Dec 29 2013