A326013 a(n) = (n+1) * (3^n + 1)^n.

1, 8, 300, 87808, 226060880, 5189199673344, 1059339584023000000, 1920530508944313150734336, 30940855519482868628911503608064, 4436292850480330161986749628748086640640, 5670112877213289113061309258575292968750000000000, 64696388003897763308476146732369244462492297181932629786624

Offset

0,2

Comments

More generally, the following sums are equal:

(1) Sum_{n>=0} binomial(n+k-1, n) * r^n * (p + q^n)^n,

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

for any fixed integer k; here, k = 2 and p = 1, q = 3, r = x.

Links

Table of n, a(n) for n=0..11.

Formula

O.g.f.: Sum_{n>=0} (n+1) * (3^n + 1)^n * x^n.

O.g.f.: Sum_{n>=0} (n+1) * 3^(n^2) * x^n / (1 - 3^n*x)^(n+2).

E.g.f.: sum_{n>=0} (n+1 + 3^n*x) * 3^(n^2) * exp(3^n*x) * x^n/n!.

Example

O.g.f.: A(x) = 1 + 8*x + 300*x^2 + 87808*x^3 + 226060880*x^4 + 5189199673344*x^5 + 1059339584023000000*x^6 + 1920530508944313150734336*x^7 + ... + (n+1) * (3^n + 1)^n*x^n + ...
such that
A(x) = 1/(1 - x)^2 + 2*3*x/(1 - 3*x)^3 + 3*3^4*x^2/(1 - 3^2*x)^4 + 4*3^9*x^3/(1 - 3^3*x)^5 + 5*3^16*x^4/(1 - 3^4*x)^6 + 6*3^25*x^5/(1 - 3^5*x)^7 + 7*3^36*x^6/(1 - 3^6*x)^8 + ... (n+1)*3^(n^2)*x^n/(1 - 3^n*x)^(n+2) + ...

Prog

(PARI) {a(n) = (n+1) * (3^n + 1)^n}
for(n=0, 15, print1(a(n), ", "))
(PARI) /* O.g.f. */
{a(n) = my(A = sum(m=0, n, (m+1) * 3^(m^2) * x^m / (1 - 3^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 + 3^m*x) * 3^(m^2) * exp(3^m*x +x*O(x^n)) * x^m/m! )); n!*polcoeff(A, n)}
for(n=0, 15, print1(a(n), ", "))

Crossrefs

Cf. A202989.

Sequence in context: A202559 A232372 A264248 * A222792 A304057 A305451

Adjacent sequences: A326010 A326011 A326012 * A326014 A326015 A326016

Keyword

nonn

Author

Paul D. Hanna, Jun 05 2019

Status

approved