A326436 - OEIS

%I #7 Jul 20 2019 16:34:09

%S 1,6,95,2307,78000,3433831,188460821,12508220886,981371259995,

%T 89426179550623,9331384489007032,1102143627943740931,

%U 145924317814992561097,21480095845779426077750,3490477008130417972086807,622292123277813938275834747,121062971468108753273621477712,25577093024015935514169919403295

%N E.g.f.: exp(-5) * Sum_{n>=0} (exp(n*x) + 4)^n / n!.

%C More generally, the following sums are equal:

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

%C (2) exp(-(p+1)*r) * Sum_{n>=0} q^(n^2) * exp(p*q^n*r) * r^n / n!,

%C here, q = exp(x), p = 4, r = 1.

%F E.g.f.: exp(-5) * Sum_{n>=0} (exp(n*x) + 4)^n / n!.

%F E.g.f.: exp(-5) * Sum_{n>=0} exp(n^2*x) * exp( 4*exp(n*x) ) / n!.

%F FORMULAS FOR TERMS.

%F a(3*n+1) = 0 (mod 2), a(3*n) = 1 (mod 2), and a(3*n+2) = 1 (mod 2) for n >= 0.

%e E.g.f.: A(x) = 1 + 6*x + 95*x^2/2! + 2307*x^3/3! + 78000*x^4/4! + 3433831*x^5/5! + 188460821*x^6/6! + 12508220886*x^7/7! + 981371259995*x^8/8! + 89426179550623*x^9/9! + 9331384489007032*x^10/10! + ...

%e such that

%e A(x) = exp(-5) * (1 + (exp(x) + 4) + (exp(2*x) + 4)^2/2! + (exp(3*x) + 4)^3/3! + (exp(4*x) + 4)^4/4! + (exp(5*x) + 4)^5/5! + (exp(6*x) + 4)^6/6! + ...)

%e also

%e A(x) = exp(-5) * (exp(4) + exp(x)*exp(4*exp(x)) + exp(4*x)*exp(4*exp(2*x))/2! + exp(9*x)*exp(4*exp(3*x))/3! + exp(16*x)*exp(4*exp(4*x))/4! + exp(25*x)*exp(4*exp(5*x))/5! + exp(36*x)*exp(4*exp(6*x))/6! + ...).

%o (PARI) /* Requires suitable precision */

%o \p200

%o Vec(round(serlaplace( exp(-5) * sum(n=0, 500, (exp(n*x +O(x^31)) + 4)^n/n! ) )))

%Y Cf. A326600, A020557, A326433, A326434, A326435, A326437.

%K nonn

%O 0,2

%A _Paul D. Hanna_, Jul 11 2019