A094418 Generalized ordered Bell numbers Bo(5,n).
1, 5, 55, 905, 19855, 544505, 17919055, 687978905, 30187495855, 1490155456505, 81732269223055, 4931150091426905, 324557348772511855, 23141780973332248505, 1776997406800302687055, 146197529083891406394905, 12829862285488250150167855, 1196280147496701351115120505
Offset: 0
Keywords
Links
- Vincenzo Librandi, Table of n, a(n) for n = 0..200
- Paul Barry, Three Études on a sequence transformation pipeline, arXiv:1803.06408 [math.CO], 2018.
Programs
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Magma
A094416:= func< n,k | (&+[Factorial(j)*n^j*StirlingSecond(k,j): j in [0..k]]) >; A094418:= func< k | A094416(5,k) >; [A094418(n): n in [0..30]]; // G. C. Greubel, Jan 12 2024
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Mathematica
t = 30; Range[0, t]! CoefficientList[Series[1/(6 - 5 Exp[x]), {x, 0, t}], x] (* Vincenzo Librandi, Mar 16 2014 *) -
PARI
my(N=25,x='x+O('x^N)); Vec(serlaplace(1/(6 - 5*exp(x)))) \\ Joerg Arndt, Jan 15 2024 -
SageMath
def A094416(n,k): return sum(factorial(j)*n^j*stirling_number2(k,j) for j in range(k+1)) # array def A094418(k): return A094416(5,k) [A094418(n) for n in range(31)] # G. C. Greubel, Jan 12 2024
Formula
E.g.f.: 1/(6 - 5*exp(x)).
a(n) = Sum_{k=0..n} A131689(n,k) * 5^k. - Philippe Deléham, Nov 03 2008
a(n) ~ n! / (6*(log(6/5))^(n+1)). - Vaclav Kotesovec, Mar 14 2014
a(0) = 1; a(n) = 5 * Sum_{k=1..n} binomial(n,k) * a(n-k). - Ilya Gutkovskiy, Jan 17 2020
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
From Seiichi Manyama, Jun 01 2025: (Start)
a(n) = (-1)^(n+1)/6 * Li_{-n}(6/5), where Li_{n}(x) is the polylogarithm function.
a(n) = (1/6) * Sum_{k>=0} k^n * (5/6)^k.
a(n) = (5/6) * Sum_{k=0..n} 6^k * (-1)^(n-k) * A131689(n,k) for n > 0. (End)
Comments