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Generalized ordered Bell numbers Bo(6,n).
18

%I #38 Jan 15 2024 01:44:27

%S 1,6,78,1518,39390,1277646,49729758,2258233998,117196187550,

%T 6842432930766,443879517004638,31674687990494478,2465744921215207710,

%U 207943837884583262286,18885506918597311159518

%N Generalized ordered Bell numbers Bo(6,n).

%C Sixth row of array A094416, which has more information.

%H Vincenzo Librandi, <a href="/A094419/b094419.txt">Table of n, a(n) for n = 0..200</a>

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

%F a(n) = Sum_{k=0..n} A131689(n,k)*6^k. - _Philippe Deléham_, Nov 03 2008

%F a(n) ~ n! / (7*(log(7/6))^(n+1)). - _Vaclav Kotesovec_, Mar 14 2014

%F a(0) = 1; a(n) = 6 * Sum_{k=1..n} binomial(n,k) * a(n-k). - _Ilya Gutkovskiy_, Jan 17 2020

%F 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

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

%o (Magma)

%o A094416:= func< n,k | (&+[Factorial(j)*n^j*StirlingSecond(k,j): j in [0..k]]) >;

%o A094419:= func< k | A094416(6,k) >;

%o [A094419(n): n in [0..30]]; // _G. C. Greubel_, Jan 12 2024

%o (SageMath)

%o def A094416(n,k): return sum(factorial(j)*n^j*stirling_number2(k,j) for j in range(k+1)) # array

%o def A094419(k): return A094416(6,k)

%o [A094419(n) for n in range(31)] # _G. C. Greubel_, Jan 12 2024

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

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

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

%K nonn

%O 0,2

%A _Ralf Stephan_, May 02 2004