OFFSET
0,2
COMMENTS
The e.g.f. of A392526 equals x + x^2*A(x)^2 where A(x) is the e.g.f. of this sequence. - Paul D. Hanna, Jan 28 2026
LINKS
Paul D. Hanna, Table of n, a(n) for n = 0..300
FORMULA
a(n) = n! * Sum_{k=0..n} (2*n-k+1)^(k-1) * binomial(2*n-k+1,n-k)/k!.
E.g.f.: (1/x) * Series_Reversion( x * (exp(-x) - x) ). - Seiichi Manyama, Dec 29 2024
From Paul D. Hanna, Jan 28 2026: (Start)
E.g.f.: (1/x) * sqrt( Sum_{n>=1} d^(n-1)/dx^(n-1) LambertW(-x)^(2*n)/n! ).
E.g.f. satisfies A(x) = LambertW( -x - x^2*A(x)^2 ) / (-x). (End)
a(n) ~ sqrt(s*(1 + s + r*s^2)/(2 + r + 4*r*s + r^2*s^2)) * n^(n-1) / (exp(n) * r^n), where r = 0.1332574057588222737261878469378628593908625200845... and s = 2.065239009348607037641236295092189572191624119574... are roots of the system of equations exp(r*s)*(1 + r*s^2) = s, exp(r*s)*r*(1 + 2*s + r*s^2) = 1. - Vaclav Kotesovec, Jan 29 2026
EXAMPLE
E.g.f.: A(x) = 1 + 2*x + 15*x^2/2! + 211*x^3/3! + 4433*x^4/4! + 124741*x^5/5! + 4412815*x^6/6! + 188335981*x^7/7! + ...
MATHEMATICA
Table[n! * Sum[(2*n-k+1)^(k-1) * Binomial[2*n-k+1, n-k]/k!, {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Jan 29 2026 *)
PROG
(PARI) a(n) = n!*sum(k=0, n, (2*n-k+1)^(k-1)*binomial(2*n-k+1, n-k)/k!);
CROSSREFS
KEYWORD
nonn
AUTHOR
Seiichi Manyama, Nov 11 2024
STATUS
approved