cp's OEIS Frontend

A094418 Generalized ordered Bell numbers Bo(5,n).

%I A094418 #47 Jun 02 2025 05:29:55
%S A094418 1,5,55,905,19855,544505,17919055,687978905,30187495855,1490155456505,
%T A094418 81732269223055,4931150091426905,324557348772511855,
%U A094418 23141780973332248505,1776997406800302687055,146197529083891406394905,12829862285488250150167855,1196280147496701351115120505
%N A094418 Generalized ordered Bell numbers Bo(5,n).
%C A094418 Fifth row of array A094416, which has more information.
%H A094418 Vincenzo Librandi, <a href="/A094418/b094418.txt">Table of n, a(n) for n = 0..200</a>
%H A094418 Paul Barry, <a href="https://arxiv.org/abs/1803.06408">Three Études on a sequence transformation pipeline</a>, arXiv:1803.06408 [math.CO], 2018.
%F A094418 E.g.f.: 1/(6 - 5*exp(x)).
%F A094418 a(n) = Sum_{k=0..n} A131689(n,k) * 5^k. - _Philippe Deléham_, Nov 03 2008
%F A094418 a(n) ~ n! / (6*(log(6/5))^(n+1)). - _Vaclav Kotesovec_, Mar 14 2014
%F A094418 a(0) = 1; a(n) = 5 * Sum_{k=1..n} binomial(n,k) * a(n-k). - _Ilya Gutkovskiy_, Jan 17 2020
%F A094418 a(0) = 1; a(n) = 5 * a(n-1) - 6 * Sum_{k=1..n-1} (-1)^k * binomial(n-1,k) * a(n-k). - _Seiichi Manyama_, Nov 16 2023
%F A094418 From _Seiichi Manyama_, Jun 01 2025: (Start)
%F A094418 a(n) = (-1)^(n+1)/6 * Li_{-n}(6/5), where Li_{n}(x) is the polylogarithm function.
%F A094418 a(n) = (1/6) * Sum_{k>=0} k^n * (5/6)^k.
%F A094418 a(n) = (5/6) * Sum_{k=0..n} 6^k * (-1)^(n-k) * A131689(n,k) for n > 0. (End)
%t A094418 t = 30; Range[0, t]! CoefficientList[Series[1/(6 - 5 Exp[x]), {x, 0, t}], x] (* _Vincenzo Librandi_, Mar 16 2014 *)
%o A094418 (Magma)
%o A094418 A094416:= func< n,k | (&+[Factorial(j)*n^j*StirlingSecond(k,j): j in [0..k]]) >;
%o A094418 A094418:= func< k | A094416(5,k) >;
%o A094418 [A094418(n): n in [0..30]]; // _G. C. Greubel_, Jan 12 2024
%o A094418 (SageMath)
%o A094418 def A094416(n,k): return sum(factorial(j)*n^j*stirling_number2(k,j) for j in range(k+1)) # array
%o A094418 def A094418(k): return A094416(5,k)
%o A094418 [A094418(n) for n in range(31)] # _G. C. Greubel_, Jan 12 2024
%o A094418 (PARI) my(N=25,x='x+O('x^N)); Vec(serlaplace(1/(6 - 5*exp(x)))) \\ _Joerg Arndt_, Jan 15 2024
%Y A094418 Cf. A094416, A094417, A094419, A094422, A131689.
%Y A094418 Cf. A346984, A365568, A365569, A365570.
%K A094418 nonn
%O A094418 0,2
%A A094418 _Ralf Stephan_, May 02 2004