A353740 E.g.f. A(x) satisfies: 1 = Sum_{n>=0} 3^n * (exp(n*x) - A(x))^n / n!.

1, 1, 4, 73, 2488, 123535, 8144527, 675856090, 68118924622, 8127605321929, 1125932450595736, 178367047220336887, 31919374476052799215, 6387927327635465158768, 1417660111210685715869386, 346403593023300571689592957, 92622566428288426844609245312

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 = 3.

Links

Table of n, a(n) for n=0..16.

Formula

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

(1) 1 = Sum_{n>=0} 3^n * (exp(n*x) - A(x))^n / n!.

(2) 1 = Sum_{n>=0} 3^n * exp(n^2*x - 3*A(x)*exp(n*x)) / n!.

Example

E.g.f: A(x) = 1 + x + 4*x^2/2! + 73*x^3/3! + 2488*x^4/4! + 123535*x^5/5! + 8144527*x^6/6! + 675856090*x^7/7! + 68118924622*x^8/8! + 8127605321929*x^9/9! + ...
such that
1 = 1 + 3*(exp(x) - A(x)) + 3^2*(exp(2*x) - A(x))^2/2! + 3^3*(exp(3*x) - A(x))^3/3! + 3^4*(exp(4*x) - A(x))^4/4! + 3^5*(exp(5*x) - A(x))^5/5! + 3^6*(exp(6*x) - A(x))^6/6! + ...
also
1 = exp(-3*A(x)) + 3*exp(x - 3*A(x)*exp(x)) + 3^2*exp(4*x - 3*A(x)*exp(2*x))/2! + 3^3*exp(9*x - 3*A(x)*exp(3*x))/3! + 3^4*exp(16*x - 3*A(x)*exp(4*x))/4! + 3^5*exp(25*x - 3*A(x)*exp(5*x))/5! + ...
Related series.
log(A(x)) = x + 3*x^2/2! + 63*x^3/3! + 2190*x^4/4! + 109899*x^5/5! + 7300587*x^6/6! + 609541911*x^7/7! + 61757056830*x^8/8! + 7402438614951*x^9/9! + ...

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 * 3^m/m! ), #A-1)/3; ); H=A; n!*A[n+1]}
for(n=0, 25, print1(a(n), ", "))

Crossrefs

Cf. A326095, A353739.

Sequence in context: A089665 A092871 A222767 * A090212 A137046 A104335

Adjacent sequences: A353737 A353738 A353739 * A353741 A353742 A353743

Keyword

nonn

Author

Paul D. Hanna, May 06 2022

Status

approved