A360466 E.g.f. satisfies A(x) = exp(2 * x * exp(x) * A(x)).

1, 2, 16, 206, 3832, 93962, 2871820, 105355406, 4515648784, 221598121490, 12257187851284, 754703476252310, 51204818674338328, 3796079000648275226, 305328667748448560668, 26483633169003911205278, 2464307301750079915255840, 244872778601760932275686434

Offset

0,2

Links

Table of n, a(n) for n=0..17.

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

Formula

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

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

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

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

a(n) ~ sqrt(1+LambertW(exp(-1)/2)) * n^(n-1) / (exp(n-1) * LambertW(exp(-1)/2)^n). - Vaclav Kotesovec, Feb 08 2023

Prog

(PARI) my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(-lambertw(-2*x*exp(x)))))
(PARI) my(N=20, x='x+O('x^N)); Vec(serlaplace(-lambertw(-2*x*exp(x))/(2*x*exp(x))))
(PARI) my(N=20, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, (k+1)^(k-1)*(2*x*exp(x))^k/k!)))
(PARI) a(n) = sum(k=0, n, 2^k*k^(n-k)*(k+1)^(k-1)*binomial(n, k));

Crossrefs

Cf. A273954, A357247, A360465.

Sequence in context: A036081 A303673 A367425 * A161568 A138429 A087923

Adjacent sequences: A360463 A360464 A360465 * A360467 A360468 A360469

Keyword

nonn

Author

Seiichi Manyama, Feb 08 2023

Status

approved