cp's OEIS Frontend

E.g.f.: A(x) = 1 - LambertW(-x^2*exp(x))/x.

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

E.g.f.: A(x) = (1/x)*Series_Reversion(x/B(x)) where B(x) = 1 + x*exp(x)/B(x) = (1+sqrt(1+4*x*exp(x)))/2.

a(n) = n*A125500(n-1) for n>0, where exp(x*A(x)) = e.g.f. of A125500.

a(n) = n!*Sum_{k=0..n} C(n-k+1,k)/(n-k+1) * k^(n-k)/(n-k)!.

If A(x)^m = Sum_{n>=0} a(n,m)*x^n/n! then

a(n,m) = n!*Sum_{k=0..n} m*C(n-k+m,k)/(n-k+m) * k^(n-k)/(n-k)!.

a(n) ~ sqrt(1+LambertW(1/(2*exp(1/2)))) * n^(n-1) / (exp(n) * 2^(n+1/2) * (LambertW(1/(2*exp(1/2))))^(n+1)). - Vaclav Kotesovec, Jul 09 2013

a(n) = n! * Sum_{k=0..n} k^(n-k) * binomial(3*n-3*k+1,k)/( (3*n-3*k+1)*(n-k)! ).

a(n) = n! * Sum_{k=0..floor(n/2)} k^(n-2*k) * binomial(n-2*k+1,k)/( (n-2*k+1)*(n-2*k)! ).

From Vaclav Kotesovec, Mar 10 2024: (Start)

E.g.f.: 1 - LambertW(-exp(x)*x^3)/x.

a(n) ~ sqrt(1 + LambertW(exp(-1/3)/3)) * n^(n-1) / (exp(n) * 3^(n + 1/2) * LambertW(exp(-1/3)/3)^(n+1)). (End)

a(n) = n!*Sum_{k=0..n} C(2*(n-k+1),k)/(n-k+1) * k^(n-k)/(n-k)!.

If A(x)^m = Sum_{n>=0} a(n,m)*x^n/n! then

a(n,m) = n!*Sum_{k=0..n} m*C(2*(n-k+m),k)/(n-k+m) * k^(n-k)/(n-k)!.

E.g.f.: A(x) = (1/x)*Series_Reversion(x/B(x)) where B(x) = (1 + x*exp(x)/B(x))^2.

a(n) ~ sqrt(2*s^(3/2)*(2-5*sqrt(s)+3*s)/(2*sqrt(s)-1)) * (2*s-2*sqrt(s))^n * n^(n-1) / exp(n), where s = 3.533778497303240223520495... is the root of the equation (2-2/sqrt(s)) * log(2*(sqrt(s)-2*s+s^(3/2))) = 1. - Vaclav Kotesovec, Jan 10 2014

E.g.f.: B(x)^2, where B(x) is the e.g.f. of A364978. - Seiichi Manyama, Nov 02 2024

a(n) = n! * Sum_{k=0..n} k^(n-k) * binomial(4*n-4*k+1,k)/( (4*n-4*k+1)*(n-k)! ).