A326011 - OEIS

A326011

a(n) = (n+1) * (2^n + 1)^n.

1, 6, 75, 2916, 417605, 234812358, 527932234375, 4755738419928072, 171280331996409907209, 24606864966197875457438730, 14080929986159936046600341796875, 32073236633246852578917758577924120588, 290760173774986242601808360162358149769707533, 10492680499171055486742235424276666079725581443186702, 1507792223578968167717594884445653164343553232898773193359375

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OFFSET

0,2

COMMENTS

More generally, the following sums are equal:

(1) Sum_{n>=0} binomial(n+k-1, n) * y^n * (F + G^n)^n,

(2) Sum_{n>=0} binomial(n+k-1, n) * y^n * G^(n^2) / (1 - y*F*G^n)^(n+k),

for any fixed integer k; here, k = 2 and y = x, F = 1, G = 2.

LINKS

Table of n, a(n) for n=0..14.

FORMULA

O.g.f.: Sum_{n>=0} (n+1) * (2^n + 1)^n * x^n.

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

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

EXAMPLE

O.g.f.: A(x) = 1 + 6*x + 75*x^2 + 2916*x^3 + 417605*x^4 + 234812358*x^5 + 527932234375*x^6 + 4755738419928072*x^7 + ... + (n+1)*(2^n + 1)^n*x^n + ...

such that

A(x) = 1/(1 - x)^2 + 2*2*x/(1 - 2*x)^3 + 3*2^4*x^2/(1 - 2^2*x)^4 + 4*2^9*x^3/(1 - 2^3*x)^5 + 5*2^16*x^4/(1 - 2^4*x)^6 + 6*2^25*x^5/(1 - 2^5*x)^7 + 7*2^36*x^6/(1 - 2^6*x)^8 + ... + (n+1)*2^(n^2)*x^n/(1 - 2^n*x)^(n+2) + ...

MATHEMATICA

Table[(n+1)(2^n+1)^n, {n, 0, 20}] (* Harvey P. Dale, Mar 22 2020 *)

PROG

(PARI) {a(n) = (n+1) * (2^n + 1)^n}

for(n=0, 15, print1(a(n), ", "))

(PARI) /* O.g.f. */

{a(n) = my(A = sum(m=0, n, (m+1) * 2^(m^2) * x^m / (1 - 2^m*x +x*O(x^n))^(m+2) )); polcoeff(A, n)}

for(n=0, 15, print1(a(n), ", "))

(PARI) /* E.g.f. */

{a(n) = my(A = sum(m=0, n, (m+1 + 2^m*x) * 2^(m^2) * exp(2^m*x +x*O(x^n)) * x^m/m! )); n!*polcoeff(A, n)}

for(n=0, 15, print1(a(n), ", "))

CROSSREFS