A008546 Quintuple factorial numbers: Product_{k = 0..n-1} (5*k + 4).

1, 4, 36, 504, 9576, 229824, 6664896, 226606464, 8837652096, 388856692224, 19053977918976, 1028914807624704, 60705973649857536, 3885182313590882304, 268077579637770878976, 19837740893195045044224, 1567181530562408558493696, 131643248567242318913470464

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.

Keiichi Shigechi, On the lattice of weighted partitions, arXiv:2212.14666 [math.CO], 2022. See p. 27.

Formula

a(n) = 4*A034301(n) = (5*n - 1)(!^5), n >= 1, with a(0) = 1.

a(n) = A011801(n + 1, 1) (first column of triangle).

a(n) ~ (sqrt(2*Pi)/Gamma(4/5))*n^(n + 3/10)*(5/e)^n*(1 + 1/(300*n) + ...). - Joe Keane (jgk(AT)jgk.org), Nov 24 2001

G.f.: 1/(1 - 4*x/(1 - 5*x/(1 - 9*x/(1 - 10*x/(1 - 14*x/(1 - 15*x/(1 - 19*x/(1 - 20*x/(1 - 24*x/(1 - ... (continued fraction). - Philippe Deléham, Jan 08 2012

a(n) = (-1)^n*Sum_{k = 0..n} 5^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*(5*k - 1)/(1 - x*(5*k + 5)/Q(k + 1) ); (continued fraction); e.g.f. (1 - 5*x)^(-4/5). - Sergei N. Gladkovskii, Mar 20 2013

G.f.: 1/x - G(0)/(2*x), where G(k) = 1 + 1/(1 - x*(5*k - 1)/(x*(5*k - 1) + 1/G(k + 1))); (continued fraction). - Sergei N. Gladkovskii, May 27 2013

a(n) = 5^n * Gamma(n + 4/5) / Gamma(4/5). - Vaclav Kotesovec, Jan 28 2015

a(n) + (-5*n + 1)*a(n - 1) = 0. - R. J. Mathar, Sep 04 2016

G.f.: 1/(1 - 4*x - 20*x^2/(1 - 14*x - 90*x^2/(1 - 24*x - 210*x^2/(1 - 34*x - 380*x^2/(1 - 44*x - 600*x^2/(1 - 54*x - 870*x^2/(1 - ...))))))) (Jacobi continued fraction). - Nikolaos Pantelidis, Feb 29 2020

Sum_{n>=0} 1/a(n) = 1 + (e/5)^(1/5)*(Gamma(4/5) - Gamma(4/5, 1/5)). - Amiram Eldar, Dec 19 2022

Maple

f:= n-> product(5*k+4, k=0..n-1);

Mathematica

FoldList[Times, 1, 5Range[0, 20] + 4] (* Vincenzo Librandi, Jun 10 2013 *)
CoefficientList[Series[(1 - 5x)^(-4/5), {x, 0, 20}], x] Range[0, 20]! (* Vaclav Kotesovec, Jan 28 2015 *)
Table[5^n Pochhammer[4/5, n], {n, 0, 20}] (* G. C. Greubel, Aug 20 2019 *)

Prog

(PARI) vector(20, n, n--; prod(j=0, n-1, 5*j+4) ) \\ G. C. Greubel, Aug 20 2019
(Magma) [1] cat [(&*[5*k+4: k in [0..n-1]]): n in [1..20]]; // G. C. Greubel, Aug 20 2019
(Sage) [5^n*rising_factorial(4/5, n) for n in (0..20)] # G. C. Greubel, Aug 20 2019
(GAP) List([0..20], n-> Product([0..n-1], k-> 5*k+4 )); # G. C. Greubel, Aug 20 2019

Crossrefs

Cf. A011801, A034301, A048994.

Cf. A008548, A047055, A047056, A051150, A052562, A254287.

Sequence in context: A354264 A138435 A366337 * A277404 A024253 A052746

Adjacent sequences: A008543 A008544 A008545 * A008547 A008548 A008549

Keyword

nonn

Author

Joe Keane (jgk(AT)jgk.org)

Status

approved