A372278 E.g.f. A(x) satisfies A(x) = exp( x * (1 + A(x)^(5/2)) ).

1, 2, 14, 218, 5256, 172332, 7161964, 360849848, 21378442976, 1456505344592, 112197636802224, 9643110922761648, 914870017865191936, 94969006015521439232, 10707303771557931935744, 1302965738334245437242368, 170216425515761065556430336

Offset

0,2

Links

Table of n, a(n) for n=0..16.

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

Formula

E.g.f.: A(x) = exp( x - 2/5 * LambertW(-5*x/2 * exp(5*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.: Sum_{k>=0} (5*k/2+1)^(k-1) * x^k/(1 - (5*k/2+1)*x)^(k+1).

a(n) ~ sqrt(1 + LambertW(exp(-1))) * 5^(n-1) * n^(n-1) / (2^(n-1) * LambertW(exp(-1))^(n + 2/5) * exp(n)). - Vaclav Kotesovec, May 06 2024

Prog

(PARI) my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(x-2/5*lambertw(-5*x/2*exp(5*x/2)))))
(PARI) a(n, r=1, t=0, u=5/2) = 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(sum(k=0, N, (5*k/2+1)^(k-1)*x^k/(1-(5*k/2+1)*x)^(k+1)))

Crossrefs

Cf. A349562, A362694, A362734, A372236, A372235.

Cf. A349716, A372279.

Sequence in context: A136550 A068369 A034405 * A197210 A153668 A105749

Adjacent sequences: A372275 A372276 A372277 * A372279 A372280 A372281

Keyword

nonn

Author

Seiichi Manyama, Apr 25 2024

Status

approved