A371524 E.g.f. A(x) satisfies A(x) = exp( 2 * x * (1 + A(x)^(1/4)) ).

1, 4, 20, 124, 936, 8424, 88648, 1072432, 14702720, 225692128, 3839770656, 71780577312, 1463532416320, 32337850727680, 770039603953664, 19664621381714944, 536234348295180288, 15554459021934423552, 478297493455731968512, 15543431292269887979008

Offset

0,2

Links

Table of n, a(n) for n=0..19.

Eric Weisstein's World of Mathematics, Lambert W-Function.

Formula

E.g.f.: A(x) = exp( 2*x - 4*LambertW(-x/2 * exp(x/2)) ).

If e.g.f. satisfies A(x) = exp( r*x*A(x)^(t/r) * (1 + A(x)^(u/r)) ), then a(n) = r * Sum_{k=0..n} (t*n+u*k+r)^(n-1) * binomial(n,k).

G.f.: 2 * Sum_{k>=0} (k/2+2)^(k-1) * x^k/(1 - (k/2+2)*x)^(k+1).

a(n) ~ sqrt(LambertW(exp(-1)) + 1) * n^(n-1) / (2^(n-2) * exp(n) * LambertW(exp(-1))^(n+4)). - Vaclav Kotesovec, Apr 24 2024

Prog

(PARI) my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(2*x-4*lambertw(-x/2*exp(x/2)))))
(PARI) a(n, r=2, t=0, u=1/2) = r*sum(k=0, n, (t*n+u*k+r)^(n-1)*binomial(n, k));
(PARI) my(N=20, x='x+O('x^N)); Vec(2*sum(k=0, N, (k/2+2)^(k-1)*x^k/(1-(k/2+2)*x)^(k+1)))

Crossrefs

Cf. A007889, A372236.

Sequence in context: A067121 A002793 A162509 * A297924 A151341 A349603

Adjacent sequences: A371521 A371522 A371523 * A371525 A371526 A371527

Keyword

nonn,new

Author

Seiichi Manyama, Apr 23 2024

Status

approved