A326097 E.g.f. A(x) satisfies: 1 = Sum_{n>=0} ((1+x)^n - A(x))^n / n!.

1, 1, 1, 11, 160, 3634, 110891, 4335204, 208768568, 12053087736, 817245047097, 64036149563110, 5723761837812580, 577407946342497516, 65153800747494185897, 8160944217790837737502, 1127265018043808661117840, 170726388496282298937412944, 28207398922198230159415688865, 5061214928838269566809894806406

Offset: 0

Author

Paul D. Hanna, Jun 06 2019

Keywords

• nonn

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 = (1+x) with p = -A(x), r = 1.

Examples

			E.g.f.: A(x) = 1 + x + x^2/2! + 11*x^3/3! + 160*x^4/4! + 3634*x^5/5! + 110891*x^6/6! + 4335204*x^7/7! + 208768568*x^8/8! + 12053087736*x^9/9! + 817245047097*x^10/10! + ...
such that
1 = 1 + ((1+x) - A(x)) + ((1+x)^2 - A(x))^2/2! + ((1+x)^3 - A(x))^3/3! + ((1+x)^4 - A(x))^4/4! + ((1+x)^5 - A(x))^5/5! + ((1+x)^6 - A(x))^6/6! + ...
also
1 = exp(-A(x)) + (1+x)*exp(-(1+x)*A(x)) + (1+x)^4*exp(-(1+x)^2*A(x))/2! + (1+x)^9*exp(-(1+x)^3*A(x))/3! + (1+x)^16*exp(-(1+x)^4*A(x))/4! + (1+x)^25*exp(-(1+x)^5*A(x))/5! + ...
		

Links

• Paul D. Hanna, Table of n, a(n) for n = 0..195

Programs

PARI

{a(n) = my(A=[1]); for(i=1,n, A=concat(A,0); A[#A] = polcoeff( sum(m=0,#A, ((1+x)^m - Ser(A))^m/m! ),#A-1) ); n!*A[n+1]}
for(n=0,20,print1(a(n),", "))

Formula

E.g.f. A(x) satisfies:

(1) 1 = Sum_{n>=0} ((1+x)^n - A(x))^n / n!.

(2) 1 = Sum_{n>=0} (1+x)^(n^2) * exp(-(1+x)^n*A(x)) / n!.

a(4*n+2) = 1 (mod 2), otherwise the terms a(k) are even for k > 3 (conjecture).