A326084 E.g.f.: Sum_{n>=0} 2^(n^2) * (1+x)^(2^n) * x^n / n!.

1, 3, 24, 716, 83072, 39087872, 75401303680, 594443022065664, 19031019101108305920, 2460683831564819120390144, 1280084328364612961085149413376, 2672769650751366373008246767117336576, 22366167127368888262542197147193185822310400, 749477178086404731802603163228930825036318134763520

Offset

0,2

Comments

More generally, the following sums are equal:

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

(2) Sum_{n>=0} q^(n^2) * exp(p*q^n*r) * r^n/n!;

here, q = 2 and p = log(1+x)/x, r = x.

Links

Table of n, a(n) for n=0..13.

Formula

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

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

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

a(n) = n! * Sum_{k=0..n} 2^(k^2) * binomial(2^k,n-k) / k!.

Example

E.g.f.: A(x) = 1 + 3*x + 24*x^2/2! + 716*x^3/3! + 83072*x^4/4! + 39087872*x^5/5! + 75401303680*x^6/6! + 594443022065664*x^7/7! + 19031019101108305920*x^8/8! + ...
such that
A(x) = 1 + 2*(1+x)^2*x + 2^4*(1+x)^4*x^2/2! + 2^9*(1+x)^8*x^3/3! + 2^16*(1+x)^16*x^4/4! + 2^25*(1+x)^32*x^5/5! + 2^36*(1+x)^64*x^6/6! + ...
also
A(x) = 1 + (2*x + log(1+x)) + (2^2*x + log(1+x))^2/2! + (2^3*x + log(1+x))^3/3! + (2^4*x + log(1+x))^4/4! + (2^5*x + log(1+x))^5/5! + (2^6*x + log(1+x))^6/6! + ...

Prog

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

Crossrefs

Cf. A326085.

Sequence in context: A202944 A056207 A297561 * A301525 A277177 A226053

Adjacent sequences: A326081 A326082 A326083 * A326085 A326086 A326087

Keyword

nonn

Author

Paul D. Hanna, Jun 10 2019

Status

approved