A259208 E.g.f.: Sum_{n>=0} x^n * (1 + x^n)^n / n!.

1, 1, 3, 1, 25, 1, 721, 1, 6721, 181441, 151201, 1, 203575681, 1, 121080961, 108972864001, 3491282995201, 1, 133541574566401, 1, 304119455447808001, 212878925715456001, 309744468633601, 1, 17665560315112008499201, 646300418472124416000001, 841941782922240001

Offset: 0

Author

Paul D. Hanna, Jun 21 2015

Keywords

• nonn

Comments

More generally, the following sums are equal:

(1) Sum_{n>=0} (q^n + p)^n * r^n/n!,

(2) Sum_{n>=0} q^(n^2) * exp(p*q^n*r) * r^n/n!;

here, q = x, p = 1, r = x. - Paul D. Hanna, Jul 03 2019

In general, let F(x) be a formal power series in x such that F(0)=1, then

Sum_{n>=0} m^n * F(q^n*r)^p * log( F(q^n*r) )^n / n! =

Sum_{n>=0} r^n * [y^n] F(y)^(m*q^n + p);

here, F(x) = exp(x), q = , p = 1, r = x, m = 1. - Paul D. Hanna, Jul 06 2019

Examples

			E.g.f.: A(x) = 1 + x + 3*x^2/2! + x^3/3! + 25*x^4/4! + x^5/5! + 721*x^6/6! + x^7/7! + 6721*x^8/8! + 181441*x^9/9! +...
where
A(x) = 1 + x*(1+x) + x^2*(1+x^2)^2/2! + x^3*(1+x^3)^3/3! + x^4*(1+x^4)^4/4! + x^5*(1+x^5)^5/5! + x^6*(1+x^6)^6/6! +...
also
A(x) = exp(x) + x^2*exp(x^2) + x^6*exp(x^3)/2! + x^12*exp(x^4)/3! + x^20*exp(x^5)/4! + x^30*exp(x^6)/5! + x^42*exp(x^7)/6! +...
RELATED SERIES.
Below we illustrate the following identity at specific values of x:
Sum_{n>=0} x^n * (1 + x^n)^n / n!  =  Sum_{n>=0} x^(n*(n+1)) * exp(x^(n+1)) / n!.
(1) At x = 1/2, the following sums are equal
S1 = Sum_{n>=0} 2^(-n*(n+1)) * (2^n + 1)^n / n!,
S1 = Sum_{n>=0} 2^(-n*(n+1)) * exp( 1/2^(n+1) ) / n!,
where S1 = 1.97862370255774939923047215233920359461142155926360261512472...
(2) At x = 1/3, the following sums are equal
S2 = Sum_{n>=0} 3^(-n*(n+1)) * (3^n + 1)^n / n!,
S2 = Sum_{n>=0} 3^(-n*(n+1)) * exp( 1/3^(n+1) ) / n!,
where S2 = 1.52049327799122758174016893855657751957768595647287750170026...
		

Links

• Paul D. Hanna, Table of n, a(n) for n = 0..1025

Crossrefs

Cf. A326096.

Programs

PARI

{a(n) = local(A=1); A = sum(m=0, n, x^m/m!*(1 + x^m +x*O(x^n))^m); n!*polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))

PARI

{a(n) = local(A=1); A = sum(m=0, n, x^(m*(m+1)) / m! * exp(x^(m+1) +x*O(x^n))); n!*polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))

PARI

{a(n) = if(n==0,1, sumdiv(n,d, binomial(d, n/d-1) * n!/d! ) )}
for(n=0, 30, print1(a(n), ", "))

Formula

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

a(n) = Sum_{d|n} binomial(d, n/d - 1) * n!/d!.