A094419 Generalized ordered Bell numbers Bo(6,n).

1, 6, 78, 1518, 39390, 1277646, 49729758, 2258233998, 117196187550, 6842432930766, 443879517004638, 31674687990494478, 2465744921215207710, 207943837884583262286, 18885506918597311159518, 1837699347783655374914958, 190743171535070652261555870, 21035482423625416328497024206

Offset: 0

Author

Ralf Stephan, May 02 2004

Keywords

• nonn

Comments

Sixth row of array A094416, which has more information.

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..200
• Wikipedia, Polylogarithm.

Crossrefs

Cf. A032033, A094416, A094417, A094418, A131689.

Cf. A346985, A354252, A365555, A365556, A365557.

Cf. A384514, A384521, A384522, A384523.

Programs

Magma

A094416:= func< n,k | (&+[Factorial(j)*n^j*StirlingSecond(k,j): j in [0..k]]) >;
A094419:= func< k | A094416(6,k) >;
[A094419(n): n in [0..30]]; // G. C. Greubel, Jan 12 2024

Mathematica

t = 30; Range[0, t]! CoefficientList[Series[1/(7 - 6 Exp[x]),{x, 0, t}], x] (* Vincenzo Librandi, Mar 16 2014 *)

PARI

my(N=25,x='x+O('x^N)); Vec(serlaplace(1/(7-6*exp(x)))) \\ Joerg Arndt, Jan 15 2024

PARI

a(n) = (-1)^(n+1)*polylog(-n, 7/6)/7; \\ Seiichi Manyama, Jun 01 2025

SageMath

def A094416(n,k): return sum(factorial(j)*n^j*stirling_number2(k,j) for j in range(k+1)) # array
def A094419(k): return A094416(6,k)
[A094419(n) for n in range(31)] # G. C. Greubel, Jan 12 2024

Formula

E.g.f.: 1/(7 - 6*exp(x)).

a(n) = Sum_{k=0..n} A131689(n,k) * 6^k. - Philippe Deléham, Nov 03 2008

a(n) ~ n! / (7*(log(7/6))^(n+1)). - Vaclav Kotesovec, Mar 14 2014

a(0) = 1; a(n) = 6 * Sum_{k=1..n} binomial(n,k) * a(n-k). - Ilya Gutkovskiy, Jan 17 2020

a(0) = 1; a(n) = 6 * a(n-1) - 7 * Sum_{k=1..n-1} (-1)^k * binomial(n-1,k) * a(n-k). - Seiichi Manyama, Nov 17 2023

From Seiichi Manyama, Jun 01 2025: (Start)

a(n) = (-1)^(n+1)/7 * Li_{-n}(7/6), where Li_{n}(x) is the polylogarithm function.

a(n) = (1/7) * Sum_{k>=0} k^n * (6/7)^k.

a(n) = (6/7) * Sum_{k=0..n} 7^k * (-1)^(n-k) * A131689(n,k) for n > 0. (End)