A202989 E.g.f: Sum_{n>=0} 3^(n^2) * exp(3^n*x) * x^n/n!.

1, 4, 100, 21952, 45212176, 864866612224, 151334226289000000, 240066313618039143841792, 3437872835498096514323500400896, 443629285048033016198674962874808664064, 515464807019389919369209932597753906250000000000

Offset

0,2

Comments

E.g.f. series identity: Sum_{n>=0} m^n * q^(n^2) * exp(b*q^n*x) * x^n/n! = Sum_{n>=0} (m*q^n + b)^n * x^n/n! for all q, m, b.

O.g.f. series identity: Sum_{n>=0} m^n * q^(n^2) * x^n/(1-b*q^n*x)^(n+1) = Sum_{n>=0} (m*q^n + b)^n * x^n for all q, m, b.

Links

Table of n, a(n) for n=0..10.

Formula

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

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

Example

E.g.f.: A(x) = 1 + 4*x + 100*x^2/2! + 21952*x^3/3! + 45212176*x^4/4! +..
By the series identity, the e.g.f.:
A(x) = exp(x) + 3*exp(3*x)*x + 3^4*exp(3^2*x)*x^2/2! + 3^9*exp(3^3*x)*x^3/3! +...
expands into:
A(x) = 1 + 4*x + 10^2*x^2/2! + 28^3*x^3/3! + 82^4*x^4/4! + 244^5*x^5/5! +...+ (3^n+1)^n*x^n/n! +...

Prog

(PARI) {a(n, q=3, m=1, b=1)=(m*q^n + b)^n}
(PARI) {a(n, q=3, m=1, b=1)=n!*polcoeff(sum(k=0, n, m^k*q^(k^2)*exp(b*q^k*x+x*O(x^n))*x^k/k!), n)}
(PARI) {a(n, q=3, m=1, b=1)=polcoeff(sum(k=0, n, m^k*q^(k^2)*x^k/(1-b*q^k*x +x*O(x^n))^(k+1)), n)}

Crossrefs

Cf. A165327, A060613, A244004, A136516, A326013.

Sequence in context: A198278 A202990 A324096 * A343183 A251361 A260025

Adjacent sequences: A202986 A202987 A202988 * A202990 A202991 A202992

Keyword

nonn

Author

Paul D. Hanna, Dec 26 2011

Status

approved