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

0,2

Links

Table of n, a(n) for n=0..26.

Formula

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

Example

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 +...

Prog

(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), ", "))

Crossrefs

Cf. A105795.

Sequence in context: A126221 A107086 A294790 * A089846 A258450 A131868

Adjacent sequences: A234640 A234641 A234642 * A234644 A234645 A234646

Keyword

nonn

Author

Paul D. Hanna, Dec 29 2013

Status

approved