A326092 E.g.f.: Sum_{n>=0} ((1+x)^n + 2)^n * x^n/n!.

1, 3, 11, 63, 525, 5883, 84519, 1494783, 31854489, 800205075, 23315862339, 777867156927, 29384670476709, 1245177345486987, 58718905551858015, 3060140159517853887, 175176443950054714161, 10955959246057628397987, 745058168844977314910331, 54857350105041217492956735, 4356213264604432880789346621

Offset: 0

Author

Paul D. Hanna, Jun 21 2019

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 = (1+x) and p = 2, r = x.

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 = 1+x, p = 2, r = x, m = 1.

Examples

			E.g.f.: A(x) = 1 + 3*x + 11*x^2/2! + 63*x^3/3! + 525*x^4/4! + 5883*x^5/5! + 84519*x^6/6! + 1494783*x^7/7! + 31854489*x^8/8! + 800205075*x^9/9! + 23315862339*x^10/10! + ...
such that
A(x) = 1 + ((1+x) + 2)*x + ((1+x)^2 + 2)^2*x^2/2! + ((1+x)^3 + 2)^3*x^3/3! + ((1+x)^4 + 2)^4*x^4/4! + ((1+x)^5 + 2)^5*x^5/5! + ((1+x)^6 + 2)^6*x^6/6! + ((1+x)^7 + 2)^7*x^7/7! + ...
also
A(x) = 1 + (1+x)*exp(2*x*(1+x))*x + (1+x)^4*exp(2*x*(1+x)^2)*x^2/2! + (1+x)^9*exp(2*x*(1+x)^3)*x^3/3! + (1+x)^16*exp(2*x*(1+x)^4)*x^4/4! + (1+x)^25*exp(2*x*(1+x)^5)*x^5/5! + (1+x)^36*exp(2*x*(1+x)^6)*x^6/6! + ...
		

Links

• Paul D. Hanna, Table of n, a(n) for n = 0..300

Crossrefs

Cf. A326096, A326093, A326094.

Cf. A326272.

Programs

PARI

/* E.g.f.: Sum_{n>=0} ((1+x)^n + 2)^n * x^n/n! */
{a(n) = my(A = sum(m=0,n, ((1+x)^m + 2 +x*O(x^n))^m * x^m/m! )); n!*polcoeff(A,n)}
for(n=0,25, print1(a(n),", "))

PARI

/* E.g.f.: Sum_{n>=0} (1+x)^(n^2) * exp(2*x*(1+x)^n) * x^n/n! */
{a(n) = my(A = sum(m=0,n, (1+x +x*O(x^n))^(m^2) * exp(2*x*(1+x)^m +x*O(x^n)) * x^m/m! )); n!*polcoeff(A,n)}
for(n=0,25, print1(a(n),", "))

Formula

E.g.f.: Sum_{n>=0} ((1+x)^n + 2)^n * x^n/n!,

E.g.f.: Sum_{n>=0} (1+x)^(n^2) * exp(2*x*(1+x)^n) * x^n/n!.

a(n) = 0 (mod 3) for n > 2.