A376576 E.g.f. A(x) satisfies A(x) = exp(x*A(x)/(1 - x^3)).

1, 1, 3, 16, 149, 1656, 22567, 372184, 7141689, 156630448, 3871782251, 106504501104, 3227742350197, 106879926110296, 3839600650843791, 148746681984864856, 6181806007303273073, 274355581868776940256, 12951023558423725477459, 647956009961120527442272

Offset

0,3

Links

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

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

Formula

a(n) = n! * Sum_{k=0..floor(n/3)} (n-3*k+1)^(n-3*k-1) * binomial(n-2*k-1,k)/(n-3*k)!.

E.g.f.: exp( -LambertW(-x/(1-x^3)) ).

From Vaclav Kotesovec, Oct 10 2024: (Start)

E.g.f.: -LambertW(-x/(1-x^3))*(1-x^3)/x.

a(n) ~ sqrt(2^(2/3) * 3^(5/3) / ((2*(9 + sqrt(81 + 12*exp(3))))^(1/3) - 2*exp(1)*(3/(9 + sqrt(81 + 12*exp(3))))^(1/3)) - 2*exp(1)) * 2^(2*n/3) * 3^(4*n/3) * ((9 + sqrt(81 + 12*exp(3)))^(1/3) / (2^(1/3) * (3*(9 + sqrt(81 + 12*exp(3))))^(2/3) - 6*exp(1)))^n * n^(n-1) / exp(n - 1/2). (End)

Prog

(PARI) a(n) = n!*sum(k=0, n\3, (n-3*k+1)^(n-3*k-1)*binomial(n-2*k-1, k)/(n-3*k)!);
(PARI) my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(-lambertw(-x/(1-x^3)))))

Crossrefs

Cf. A052868, A376575.

Cf. A293493.

Sequence in context: A264660 A376578 A362655 * A368451 A214933 A376565

Adjacent sequences: A376573 A376574 A376575 * A376577 A376578 A376579

Keyword

nonn

Author

Seiichi Manyama, Sep 28 2024

Status

approved