A360548 -id:A360548 - cp's OEIS frontend

A360545 E.g.f. satisfies A(x) = x * exp( 3*(x + A(x))/2 ).

0, 1, 6, 54, 756, 14580, 358668, 10736712, 378823392, 15395255280, 708217959600, 36380741745744, 2064234271203360, 128214974795177088, 8652900673357097472, 630483717450225530880, 49330027417316557012992, 4124992361928178722764544

Offset: 0

Seiichi Manyama, Feb 11 2023

nonn

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

Cf. A100526, A216857, A360548.

Cf. A360482, A360544.

my(N=20, x='x+O('x^N)); concat(0, Vec(serlaplace(-2*lambertw(-3*x/2*exp(3*x/2))/3)))

a(n) = sum(k=1, n, (3*k/2)^(n-1)*binomial(n, k));

E.g.f.: A(x) = (-2/3) * LambertW(-3*x/2 * exp(3*x/2)).
a(n) = Sum_{k=1..n} (3*k/2)^(n-1) * binomial(n,k) = 3^(n-1) * A100526(n).
a(n) ~ sqrt(1 + LambertW(exp(-1))) * 3^(n-1) * n^(n-1) / (2^(n-1) * exp(n) * LambertW(exp(-1))^n). - Vaclav Kotesovec, Feb 17 2023

A372333 Expansion of e.g.f. -exp(x) * LambertW(-2*x)/2.

Original entry on oeis.org

0, 1, 6, 51, 684, 12965, 317298, 9500631, 336237016, 13729172553, 635237632350, 32844916975739, 1876755685038468, 117437155609780461, 7986793018367861194, 586578825469711599135, 46268265552518066488752, 3901008402618593931019409

Offset: 0

Seiichi Manyama, Apr 28 2024

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..352
Eric Weisstein's World of Mathematics, Lambert W-Function.

Cf. A277473, A372334.

Cf. A360548, A372315.

my(N=20, x='x+O('x^N)); concat(0, Vec(serlaplace(-exp(x)*lambertw(-2*x)/2)))

a(n) = sum(k=1, n, (2*k)^(k-1)*binomial(n, k));

a(n) = Sum_{k=1..n} (2*k)^(k-1) * binomial(n,k).
G.f.: Sum_{k>=1} (2*k)^(k-1) * x^k / (1-x)^(k+1).
a(n) ~ exp(exp(-1)/2) * 2^(n-1) * n^(n-1). - Vaclav Kotesovec, Apr 30 2024

cp's OEIS Frontend

A360545 E.g.f. satisfies A(x) = x * exp( 3*(x + A(x))/2 ).

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

PARI

Formula

A372333 Expansion of e.g.f. -exp(x) * LambertW(-2*x)/2.

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

PARI

Formula