A008543 Sextuple factorial numbers: Product_{k=0..n-1} (6*k + 5).

1, 5, 55, 935, 21505, 623645, 21827575, 894930575, 42061737025, 2229272062325, 131527051677175, 8549258359016375, 606997343490162625, 46738795448742522125, 3879320022245629336375, 345259481979861010937375, 32799650788086796039050625, 3312764729596766399944113125

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

a(n) = A013988(n+1, 1) (first column of triangle).

Cf. A004994, A034787, A049308, A047058, A048994, A051151.

Sequence in context: A094418 A365588 A367164 * A057130 A141357 A357394

Adjacent sequences: A008540 A008541 A008542 * A008544 A008545 A008546

Keyword

nonn,easy

Author

Joe Keane (jgk(AT)jgk.org)

Status

approved