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A326014 E.g.f. A(x) satisfies: 1 = Sum_{n>=0} (exp(n*x) - A(x))^n * x^n/n!. 1
1, 1, 1, 4, 37, 376, 5851, 116348, 2654233, 73034272, 2356014871, 86009667052, 3547332820549, 164187709367696, 8421668696488867, 475395141424459636, 29395941256967070385, 1979988614183567826496, 144537864622993377619759, 11390224711130862102597980, 965536276617554529080723101, 87743847735547289888220490096 (list; graph; refs; listen; history; text; internal format)
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

Sequence in context: A199690 A133462 A123762 * A220914 A070768 A221630

Adjacent sequences:  A326011 A326012 A326013 * A326015 A326016 A326017

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Jun 05 2019

STATUS

approved

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Last modified December 4 13:29 EST 2021. Contains 349526 sequences. (Running on oeis4.)