A378041 E.g.f. satisfies A(x) = exp( x * A(x)^2 / (1-x) ) / (1-x).
1, 2, 15, 220, 4873, 145446, 5479639, 249736936, 13366083889, 821950542730, 57117681241471, 4426656694204020, 378577567656396409, 35416929943920575662, 3598006167290727776263, 394460149364865110384896, 46420283015545052734709473, 5836509710708683465245181458
Offset: 0
Keywords
Links
- Eric Weisstein's World of Mathematics, Lambert W-Function.
Programs
-
PARI
my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(-lambertw(-2*x/(1-x)^3)/2)/(1-x))) -
PARI
a(n) = n!*sum(k=0, n, (2*k+1)^(k-1)*binomial(n+2*k, n-k)/k!);
Formula
E.g.f.: exp( -LambertW(-2*x/(1-x)^3)/2 )/(1-x).
a(n) = n! * Sum_{k=0..n} (2*k+1)^(k-1) * binomial(n+2*k,n-k)/k!.
a(n) ~ 3^(n + 5/3) * c^((n + 2)/3) * n^(n-1) / (exp(n) * (3*c^(1/3) - c^(2/3)*3^(1/3) * exp(1/3) + 2*3^(2/3) * exp(2/3))^n) / (sqrt(2) * (c^(2/3) - 2*3^(1/3) * exp(1/3))^(5/2) * sqrt((3^(2/3)*c^(2/3) - 6*exp(1/3)) / (9*3^(1/3)*c^(2/3) - 8*3^(1/3)*c^(2/3) * exp(1) + 8*3^(2/3)*exp(4/3) - 15*3^(1/6) * exp(1/3)*(c/sqrt(3)) + 2*c^(1/3)*exp(2/3) * (c + 15)))), where c = 9 + sqrt(81 + 24*exp(1)). - Vaclav Kotesovec, Nov 15 2024