A326084 - OEIS

%I #10 Jun 10 2019 21:19:50

%S 1,3,24,716,83072,39087872,75401303680,594443022065664,

%T 19031019101108305920,2460683831564819120390144,

%U 1280084328364612961085149413376,2672769650751366373008246767117336576,22366167127368888262542197147193185822310400,749477178086404731802603163228930825036318134763520

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

%C More generally, the following sums are equal:

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

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

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

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

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

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

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

%e 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! + ...

%e such that

%e 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! + ...

%e also

%e 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! + ...

%o (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)}

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

%o (PARI) {a(n) = n! * sum(k=0,n, 2^(k^2) * binomial(2^k,n-k)/k!)}

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

%Y Cf. A326085.

%K nonn

%O 0,2

%A _Paul D. Hanna_, Jun 10 2019