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

%I A094419 #54 Jun 01 2025 16:38:40
%S A094419 1,6,78,1518,39390,1277646,49729758,2258233998,117196187550,
%T A094419 6842432930766,443879517004638,31674687990494478,2465744921215207710,
%U A094419 207943837884583262286,18885506918597311159518,1837699347783655374914958,190743171535070652261555870,21035482423625416328497024206
%N A094419 Generalized ordered Bell numbers Bo(6,n).
%C A094419 Sixth row of array A094416, which has more information.
%H A094419 Vincenzo Librandi, <a href="/A094419/b094419.txt">Table of n, a(n) for n = 0..200</a>
%H A094419 Wikipedia, <a href="https://en.wikipedia.org/wiki/Polylogarithm">Polylogarithm</a>.
%F A094419 E.g.f.: 1/(7 - 6*exp(x)).
%F A094419 a(n) = Sum_{k=0..n} A131689(n,k) * 6^k. - _Philippe Deléham_, Nov 03 2008
%F A094419 a(n) ~ n! / (7*(log(7/6))^(n+1)). - _Vaclav Kotesovec_, Mar 14 2014
%F A094419 a(0) = 1; a(n) = 6 * Sum_{k=1..n} binomial(n,k) * a(n-k). - _Ilya Gutkovskiy_, Jan 17 2020
%F A094419 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
%F A094419 From _Seiichi Manyama_, Jun 01 2025: (Start)
%F A094419 a(n) = (-1)^(n+1)/7 * Li_{-n}(7/6), where Li_{n}(x) is the polylogarithm function.
%F A094419 a(n) = (1/7) * Sum_{k>=0} k^n * (6/7)^k.
%F A094419 a(n) = (6/7) * Sum_{k=0..n} 7^k * (-1)^(n-k) * A131689(n,k) for n > 0. (End)
%t A094419 t = 30; Range[0, t]! CoefficientList[Series[1/(7 - 6 Exp[x]),{x, 0, t}], x] (* _Vincenzo Librandi_, Mar 16 2014 *)
%o A094419 (Magma)
%o A094419 A094416:= func< n,k | (&+[Factorial(j)*n^j*StirlingSecond(k,j): j in [0..k]]) >;
%o A094419 A094419:= func< k | A094416(6,k) >;
%o A094419 [A094419(n): n in [0..30]]; // _G. C. Greubel_, Jan 12 2024
%o A094419 (SageMath)
%o A094419 def A094416(n,k): return sum(factorial(j)*n^j*stirling_number2(k,j) for j in range(k+1)) # array
%o A094419 def A094419(k): return A094416(6,k)
%o A094419 [A094419(n) for n in range(31)] # _G. C. Greubel_, Jan 12 2024
%o A094419 (PARI) my(N=25,x='x+O('x^N)); Vec(serlaplace(1/(7-6*exp(x)))) \\ _Joerg Arndt_, Jan 15 2024
%o A094419 (PARI) a(n) = (-1)^(n+1)*polylog(-n, 7/6)/7; \\ _Seiichi Manyama_, Jun 01 2025
%Y A094419 Cf. A032033, A094416, A094417, A094418, A131689.
%Y A094419 Cf. A346985, A354252, A365555, A365556, A365557.
%Y A094419 Cf. A384514, A384521, A384522, A384523.
%K A094419 nonn
%O A094419 0,2
%A A094419 _Ralf Stephan_, May 02 2004