OFFSET
0,2
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..300
Wolfdieter Lang, On generalizations of Stirling number triangles, J. Integer Seqs., Vol. 3 (2000), Article 00.2.4.
FORMULA
a(n) = 5*A034787(n) = (6*n-1)(!^6), n >= 1, a(0) := 1.
E.g.f.: (1 - 6*x)^(-5/6).
a(n) ~ 2^(1/2)*Pi^(1/2)*Gamma(5/6)^-1*n^(1/3)*6^n*e^-n*n^n*(1 + (1/72)*n^-1 + ...). - Joe Keane (jgk(AT)jgk.org), Nov 24 2001
G.f.: 1/(1-5x/(1-6x/(1-11x/(1-12x/(1-17x/(1-18x/(1-23x/(1-24x/(1-... (continued fraction). - Philippe Deléham, Jan 08 2012
a(n) = (-1)^n*Sum_{k=0..n} 6^k*s(n+1,n+1-k), where s(n,k) are the Stirling numbers of the first kind, A048994. - Mircea Merca, May 03 2012
G.f.: (1 - 1/Q(0))/x where Q(k) = 1 - x*(6*k-1)/(1 - x*(6*k+6)/Q(k+1)); (continued fraction). - Sergei N. Gladkovskii, Mar 20 2013
a(n) = 6^n * Gamma(n+5/6) / Gamma(5/6). - Vaclav Kotesovec, Jan 28 2015
D-finite with recurrence: a(n) +(-6*n+1)*a(n-1)=0. - R. J. Mathar, Jan 17 2020
From Nikolaos Pantelidis, May 22 2022: (Start)
G.f.: 1/G(0), where G(k) = 1 - (12*k+5)*x - 6*(k+1)*(6*k+5)*x^2/G(k+1) (a continued fraction);
which starts 1/(1-5*x-30*x^2/(1-17*x-132*x^2/(1-29*x-306*x^2/(1-41*x-552*x^2/(1-53*x-870*x^2/(1-65*x-1260*x^2/(1-...))))))) (a Jacobi continued fraction).
(End)
Sum_{n>=0} 1/a(n) = 1 + (e/6)^(1/6)*(Gamma(5/6) - Gamma(5/6, 1/6)). - Amiram Eldar, Dec 18 2022
MAPLE
f := n->product( (6*k-1), k=0..n);
MATHEMATICA
FoldList[Times, 1, 6Range[0, 15]+5] (* Harvey P. Dale, Feb 20 2011 *)
Table[6^n*Pochhammer[5/6, n], {n, 0, 20}] (* G. C. Greubel, Dec 03 2019 *)
CoefficientList[Series[(1 - 6x)^(-5/6), {x, 0, 20}], x] Range[0, 20]! (* Nikolaos Pantelidis, Jan 31 2023 *)
PROG
(PARI) a(n)=prod(k=1, n, 6*k-1) \\ Charles R Greathouse IV, Aug 17 2011
(Magma) [Round(6^n*Gamma(n+5/6)/Gamma(5/6)): n in [0..20]]; // G. C. Greubel, Dec 03 2019
(Sage) [6^n*rising_factorial(5/6, n) for n in (0..20)] # G. C. Greubel, Dec 03 2019
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Joe Keane (jgk(AT)jgk.org)
STATUS
approved