OFFSET
0,4
COMMENTS
More generally, the following sums are equal:
(1) Sum_{n>=0} (q^n + p)^n * x^n/n!,
(2) Sum_{n>=0} q^(n^2) * exp(p*q^n*x) * x^n/n!;
here, q = exp(x) and p = -A(x).
LINKS
Paul D. Hanna, Table of n, a(n) for n = 0..200
FORMULA
E.g.f. A(x) = Sum_{n>=0} a(n)*x^n/n! satisfies:
(1) 1 = Sum_{n>=0} (exp(n*x) - A(x))^n * x^n/n!,
(2) 1 = Sum_{n>=0} exp(n^2*x - x*exp(n*x)*A(x)) * x^n/n!.
EXAMPLE
E.g.f.: A(x) = 1 + x + x^2/2! + 4*x^3/3! + 37*x^4/4! + 376*x^5/5! + 5851*x^6/6! + 116348*x^7/7! + 2654233*x^8/8! + 73034272*x^9/9! + 2356014871*x^10/10! + 86009667052*x^11/11! + 3547332820549*x^12/12! + 164187709367696*x^13/13! + 8421668696488867*x^14/14! + 475395141424459636*x^15/15! + ...
such that
1 = 1 + (exp(x) - A(x))*x + (exp(2*x) - A(x))^2*x^2/2! + (exp(3*x) - A(x))^3*x^3/3! + (exp(4*x) - A(x))^4*x^4/4! + (exp(5*x) - A(x))^5*x^5/5! + (exp(6*x) - A(x))^6*x^6/6! + (exp(7*x) - A(x))^7*x^7/7! + ...
also,
1 = exp(-x*A(x)) + exp(x - x*exp(x)*A(x))*x + exp(4*x - x*exp(2*x)*A(x))*x^2/2! + exp(9*x - x*exp(3*x)*A(x))*x^3/3! + exp(16*x - x*exp(4*x)*A(x))*x^4/4! + exp(25*x - x*exp(5*x)*A(x))*x^5/5! + exp(36*x - x*exp(6*x)*A(x))*x^6/6! + exp(49*x - x*exp(7*x)*A(x))*x^7/7! + exp(64*x - x*exp(8*x)*A(x))*x^8/8! + ...
PROG
(PARI) /* 1 = Sum_{n>=0} (exp(n*x) - A(x))^n * x^n/n! */
{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 * x^m/m!), #A) ); n!*A[n+1]}
for(n=0, 25, print1(a(n), ", "))
(PARI) /* 1 = Sum_{n>=0} exp(n^2*x - x*exp(n*x)*A(x)) * x^n/n! */
{a(n) = my(A=[1]); for(i=1, n, A=concat(A, 0); A[#A] = polcoeff( sum(m=0, #A, exp(m^2*x - x*exp(m*x +x*O(x^#A))*Ser(A)) * x^m/m!), #A) ); n!*A[n+1]}
for(n=0, 25, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Jun 05 2019
STATUS
approved