A372163 E.g.f. A(x) satisfies A(x) = exp( x * (1+3*x)^(1/3) * A(x) ).

1, 1, 5, 28, 321, 3636, 65947, 1154238, 28622001, 684987400, 21513702771, 656705784714, 24936869827465, 928288327257084, 41315505985090443, 1817727059210127286, 92749773791662574433, 4712674616532693996432, 271157923143678988333027

Offset

0,3

Links

Table of n, a(n) for n=0..18.

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

Formula

E.g.f.: A(x) = exp( -LambertW(-x * (1+3*x)^(1/3)) ).

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

a(n) ~ sqrt(4*r+1) * n^(n-1) / (exp(n - 5/2) * r^(n - 3/2)), where r = 0.29742497866288781360719311656731644994668261137281157848090655000... is the root of the equation r*(1 + 3*r)^(1/3) = exp(-1). - Vaclav Kotesovec, Apr 22 2024

Prog

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

Crossrefs

Cf. A362771, A372162.

Sequence in context: A057792 A338661 A174464 * A354897 A345632 A024068

Adjacent sequences: A372160 A372161 A372162 * A372164 A372165 A372166

Keyword

nonn

Author

Seiichi Manyama, Apr 20 2024

Status

approved