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.

1, 2, 5, 13, 35, 99, 297, 951, 3265, 12047, 47761, 202975, 921281, 4447327, 22737537, 122639583, 695404929, 4132531679, 25667031937, 166211936735, 1119791799425, 7833568488415, 56802921911681, 426267651506655, 3305731721387649, 26457699508131807, 218276886237532033

Offset: 0

Author

Paul D. Hanna, Dec 29 2013

Keywords

• nonn

Examples

			E.g.f.: A(x) = 1 + 2*x + 5*x^2/2! + 13*x^3/3! + 35*x^4/4! + 99*x^5/5! +...
where the e.g.f. may be expressed as a series involving iterated integration:
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 +...
		

Crossrefs

Cf. A105795.

Programs

PARI

{a(n)=sum(k=0,n, sum(j=0,k,binomial(k,j)*j^(n-k)))}
for(n=0,30,print1(a(n),", "))

PARI

{INTEGRATE(n,F)=local(G=F);for(i=1,n,G=intformal(G));G}
{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)}
for(n=0,30,print1(a(n),", "))

Formula

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