OFFSET
0,3
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 = exp(x) with p = -A(x), r = 1.
LINKS
Paul D. Hanna, Table of n, a(n) for n = 0..195
FORMULA
E.g.f. A(x) satisfies:
(1) 1 = Sum_{n>=0} (exp(n*x) - A(x))^n / n!.
(2) 1 = Sum_{n>=0} exp(n^2*x - A(x)*exp(n*x)) / n!.
EXAMPLE
E.g.f.: A(x) = 1 + x + 2*x^2/2! + 15*x^3/3! + 234*x^4/4! + 5525*x^5/5! + 176823*x^6/6! + 7232050*x^7/7! + 363749900*x^8/8! + 21891574683*x^9/9! + 1544392825386*x^10/10! + ...
such that
1 = 1 + (exp(x) - A(x)) + (exp(2*x) - A(x))^2/2! + (exp(3*x) - A(x))^3/3! + (exp(4*x) - A(x))^4/4! + (exp(5*x) - A(x))^5/5! + (exp(6*x) - A(x))^6/6! + ...
also
1 = exp(-A(x)) + exp(x - A(x)*exp(x)) + exp(4*x - A(x)*exp(2*x))/2! + exp(9*x - A(x)*exp(3*x))/3! + exp(16*x - A(x)*exp(4*x))/4! + exp(25*x - A(x)*exp(5*x))/5! + exp(36*x - A(x)*exp(6*x))/6! + ...
PROG
(PARI) {a(n) = my(A=[1]); for(i=1, n, A=concat(A, 0);
A[#A] = polcoeff( sum(m=0, #A, (exp(m*x +x*O(x^#A)) - Ser(A))^m/m! ), #A-1); ); n!*A[n+1]}
for(n=0, 25, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Jun 06 2019
STATUS
approved