A354259 -id:A354259 - cp's OEIS frontend

A354260 Expansion of e.g.f. 1/sqrt(1 - 8 * log(1+x)).

1, 4, 44, 824, 21624, 730176, 30144192, 1470979968, 82833047424, 5286741547008, 377135779749888, 29736359948175360, 2568013599548037120, 241061197802997288960, 24439230397588083240960, 2661258811775918180474880, 309780832909692738794987520

Offset: 0

Seiichi Manyama, May 21 2022

nonn

Cf. A320343, A354240, A354259.

Cf. A354253, A354262.

my(N=20, x='x+O('x^N)); Vec(serlaplace(1/sqrt(1-8*log(1+x))))

my(N=20, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, binomial(2*k, k)*(2*log(1+x))^k)))

a(n) = sum(k=0, n, 2^k*(2*k)!*stirling(n, k, 1)/k!);

E.g.f.: Sum_{k>=0} binomial(2*k,k) * (2 * log(1+x))^k.
a(n) = Sum_{k=0..n} 2^k * (2*k)! * Stirling1(n,k)/k!.
a(n) ~ n^n / (2 * (exp(1/8)-1)^(n + 1/2) * exp(n - 1/16)). - Vaclav Kotesovec, Jun 04 2022

A354261 Expansion of e.g.f. 1/sqrt(1 + 6 * log(1-x)).

Original entry on oeis.org

1, 3, 30, 492, 11250, 330282, 11844288, 501822108, 24527880756, 1358556883308, 84094256900232, 5753027212816320, 431039748845205000, 35102411472973316040, 3087236653107610062240, 291627772873980244894800, 29447260745861893561906320

Offset: 0

Seiichi Manyama, May 21 2022

nonn

Cf. A346978, A354241, A354262.

Cf. A354252, A354259.

my(N=20, x='x+O('x^N)); Vec(serlaplace(1/sqrt(1+6*log(1-x))))

my(N=20, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, binomial(2*k, k)*(-3*log(1-x)/2)^k)))

a(n) = sum(k=0, n, (3/2)^k*(2*k)!*abs(stirling(n, k, 1))/k!);

E.g.f.: Sum_{k>=0} binomial(2*k,k) * (-3 * log(1-x)/2)^k.
a(n) = Sum_{k=0..n} (3/2)^k * (2*k)! * |Stirling1(n,k)|/k!.
a(n) ~ n^n / (sqrt(3) * (exp(1/6)-1)^(n + 1/2) * exp(5*n/6)). - Vaclav Kotesovec, Jun 04 2022

cp's OEIS Frontend

A354260 Expansion of e.g.f. 1/sqrt(1 - 8 * log(1+x)).

Views

Author

Keywords

Crossrefs

Programs

PARI

PARI

PARI

Formula

A354261 Expansion of e.g.f. 1/sqrt(1 + 6 * log(1-x)).

Views

Author

Keywords

Crossrefs

Programs

PARI

PARI

PARI

Formula