A326431 E.g.f.: exp(-2) * Sum_{n>=0} ((1+x)^n + 1)^n / n!.

1, 3, 26, 393, 8806, 268011, 10496566, 509611213, 29841622422, 2063796756103, 165781539363706, 15259755609383885, 1591551797382262450, 186311156677551137459, 24281772775240615369662, 3498626608233846654660989, 553893001173840022047130286, 95833008154703833096894957199, 18033356856862268626280345672162

Offset

0,2

Comments

More generally, the following sums are equal:

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

(2) exp(-r*(p+1)) * Sum_{n>=0} q^(n^2) * exp(p*q^n*r) * r^n / n!,

here, q = 1+x, p = 1, r = 1.

Links

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

Formula

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

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

Example

E.g.f.: A(x) = 1 + 3*x + 26*x^2/2! + 393*x^3/3! + 8806*x^4/4! + 268011*x^5/5! + 10496566*x^6/6! + 509611213*x^7/7! + 29841622422*x^8/8! + 2063796756103*x^9/9! + 165781539363706*x^10/10! + ...
such that
A(x) = exp(-2) * (1 + ((1+x) + 1) + ((1+x)^2 + 1)^2/2! + ((1+x)^3 + 1)^3/3! + ((1+x)^4 + 1)^4/4! + ((1+x)^5 + 1)^5/5! + ((1+x)^6 + 1)^6/6! + ...)
also,
A(x) = exp(-2) * (exp(1) + (1+x)*exp(1+x) + (1+x)^4*exp((1+x)^2)/2! + (1+x)^9*exp((1+x)^3)/3! + (1+x)^16*exp((1+x)^4)/4! + (1+x)^25*exp((1+x)^5)/5! + (1+x)^36*exp((1+x)^6)/6! + ...).

Prog

(PARI) /* Requires appropriate precision */
\p200
{a(n) = my(A = exp(-2) * sum(m=0, n+300, ((1+x)^m + 1 +x*O(x^n))^m / m! )); round(n!*polcoeff(A, n))}
for(n=0, 20, print1(a(n), ", "))

Crossrefs

Cf. A326432.

Sequence in context: A317654 A143155 A300283 * A206403 A192554 A306280

Adjacent sequences: A326428 A326429 A326430 * A326432 A326433 A326434

Keyword

nonn

Author

Paul D. Hanna, Jul 09 2019

Status

approved