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
Wikipedia, Polylogarithm.
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
From Seiichi Manyama, Jun 01 2025: (Start)
a(n) = (-1)^(n+1)/7 * Li_{-n}(7/6), where Li_{n}(x) is the polylogarithm function.
a(n) = (1/7) * Sum_{k>=0} k^n * (6/7)^k.
a(n) = (6/7) * Sum_{k=0..n} 7^k * (-1)^(n-k) * A131689(n,k) for n > 0. (End)
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(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
[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
(PARI) a(n) = (-1)^(n+1)*polylog(-n, 7/6)/7; \\ Seiichi Manyama, Jun 01 2025
CROSSREFS
KEYWORD
nonn
AUTHOR
Ralf Stephan, May 02 2004
STATUS
approved