A337851 a(n) = (2^n + 2)^n.

1, 4, 36, 1000, 104976, 45435424, 82653950016, 627485170000000, 19631688197463081216, 2504194578379511247798784, 1292628144912333835229805413376, 2687153475176994340820312500000000000, 22431765115399782718874449007331506546282496

Offset

0,2

Comments

In general, we have the o.g.f. 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 ; here, q=2, m=1, b=2.

In general, we have the e.g.f. 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! ; here, q=2, m=1, b=2.

Links

Table of n, a(n) for n=0..12.

Formula

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

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

a(n) = 2^n * A165327(n) for n >= 0.

Example

O.g.f.: A(x) = 1 + 4*x + 36*x^2 + 1000*x^3 + 104976*x^4 + 45435424*x^5 + 82653950016*x^6 + 627485170000000*x^7 + 19631688197463081216*x^8 + ...
where
A(x) = 1/(1 - 2*x) + 2*x/(1 - 2^2*x)^2 + 2^4*x^2/(1 - 2^3*x)^3 + 2^9*x^3/(1 - 2^4*x)^4 + 2^16*x^4/(1 - 2^5*x)^5 + 2^25*x^5/(1 - 2^6*x)^6 + ...

Prog

(PARI) {a(n, q, m, b) = (m*q^n + b)^n}
for(n=0, 15, print1(a(n, q=2, m=1, b=2), ", "))
(PARI) /* E.g.f. formula: */
{a(n, q, m, b) = polcoeff( sum(k=0, n, m^k * q^(k^2) * x^k / (1 - b*q^k*x +x*O(x^n))^(k+1)), n)}
for(n=0, 15, print1(a(n, q=2, m=1, b=2), ", "))
(PARI) /* E.g.f. formula: */
{a(n, q, m, b) = n! * polcoeff( sum(k=0, n, m^k * q^(k^2) * exp(b*q^k*x +x*O(x^n)) * x^k/k!), n)}
for(n=0, 15, print1(a(n, q=2, m=1, b=2), ", "))

Crossrefs

Cf. A165327, A055601, A251657, A337852, A136516.

Sequence in context: A061742 A136469 A120605 * A353410 A360725 A173212

Adjacent sequences: A337848 A337849 A337850 * A337852 A337853 A337854

Keyword

nonn

Author

Paul D. Hanna, Sep 26 2020

Status

approved