OFFSET
0,2
COMMENTS
More generally, the following sums are equal:
(1) exp(-(p+1)*r) * Sum_{n>=0} (q^n + p)^n * r^n / n!,
(2) exp(-(p+1)*r) * Sum_{n>=0} q^(n^2) * exp(p*q^n*r) * r^n / n!,
here, q = exp(x), p = 4, r = 1.
FORMULA
E.g.f.: exp(-5) * Sum_{n>=0} (exp(n*x) + 4)^n / n!.
E.g.f.: exp(-5) * Sum_{n>=0} exp(n^2*x) * exp( 4*exp(n*x) ) / n!.
FORMULAS FOR TERMS.
a(3*n+1) = 0 (mod 2), a(3*n) = 1 (mod 2), and a(3*n+2) = 1 (mod 2) for n >= 0.
EXAMPLE
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! + ...
such that
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! + ...)
also
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! + ...).
PROG
(PARI) /* Requires suitable precision */
\p200
Vec(round(serlaplace( exp(-5) * sum(n=0, 500, (exp(n*x +O(x^31)) + 4)^n/n! ) )))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Jul 11 2019
STATUS
approved