cp's OEIS Frontend

E.g.f. A(x) = Sum_{n>=0} a(n)*x^n/n! satisfies the following formulas.

(1) A(x) = 1 + x*A(x)*exp(x*A(x)).

(2) A(x) = (1/x) * Series_Reversion( x/(1 + x*exp(x)) ).

(3) A(x) = 1 + (m+1) * Sum{n>=1} n*(n+m)^(n-2) * x^n/n! * A(x)^n * exp(-(n+m-1)*x*A(x)) for all fixed nonnegative m.

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

Let A(x)^m = Sum_{n>=0} a(n,m)*x^n/n! then a(n,m) = n! * Sum_{k=0..n} binomial(n+m,k)*m/(n+m) * k^(n-k)/(n-k)!.

a(n) ~ n^(n-1) * c * ((c-1)*c)^(n+1/2) / (sqrt(2*c-1) * exp(n)), where c = 1 + 1/(2*LambertW(1/2)) = 2.4215299358831166... - Vaclav Kotesovec, Jan 10 2014

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

a(n) ~ sqrt((1 + r*s^2)/(6 + 4*r*s^2)) * n^(n-1) / (exp(n) * r^(n + 1/2)), where r = 0.2190923703746024362724546703711998154573791458000... and s = 1.747404632046819382844696016554403302840973484745... are real roots of the system of equations 1 + exp(r*s^2)*r*s = s, 2*r*s^2*(s-1) = 1. - Vaclav Kotesovec, Nov 18 2023

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

a(n) ~ sqrt(s*(1 + 2*r*s) / (4 + 3*r - 12*r*s + 12*r^2*s^2 - 4*r^3*s^3)) * n^(n-1) / (exp(n) * r^n), where r = 0.1811100305436879929789759231994897963241226689... and s = 1.893740207738561813713992833266450862854198944672... are real roots of the system of equations exp(r/(1 - r*s)^3) = s, 3*s*r^2 = (1 - r*s)^4. - Vaclav Kotesovec, Nov 18 2023

E.g.f.: B(x)^3, where B(x) is the e.g.f. of A364981.

a(n) = 3 * n! * Sum_{k=0..n} k^(n-k) * binomial(3*n-2*k+3,k)/( (3*n-2*k+3)*(n-k)! ).

a(n) = n! * Sum_{k=0..n} k^(n-k) * binomial(4*n-3*k+1,k)/( (4*n-3*k+1)*(n-k)! ).