A326095 E.g.f. A(x) satisfies: 1 = Sum_{n>=0} (exp(n*x) - A(x))^n / n!.

1, 1, 2, 15, 234, 5525, 176823, 7232050, 363749900, 21891574683, 1544392825386, 125684334518985, 11648104664937271, 1216426938053726672, 141882106115149781072, 18344653087551340567427, 2612237303669636927142962, 407360290222179197806752141, 69221669418346150774013957483, 12760799611726977737400430776570, 2541919524892729009158279375599352

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

Sequence in context: A216860 A161968 A294043 * A212370 A302358 A192561

Adjacent sequences: A326092 A326093 A326094 * A326096 A326097 A326098

Keyword

nonn

Author

Paul D. Hanna, Jun 06 2019

Status

approved