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

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

Offset

0,2

Comments

Sixth row of array A094416, which has more information.

Links

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

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

Mathematica

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

Prog

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

Crossrefs

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

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

Sequence in context: A131926 A132866 A279444 * A229044 A300874 A049209

Adjacent sequences: A094416 A094417 A094418 * A094420 A094421 A094422

Keyword

nonn

Author

Ralf Stephan, May 02 2004

Status

approved