A008548 Quintuple factorial numbers: Product_{k=0..n-1} (5*k+1).

1, 1, 6, 66, 1056, 22176, 576576, 17873856, 643458816, 26381811456, 1213563326976, 61891729675776, 3465936861843456, 211422148572450816, 13953861805781753856, 990724188210504523776, 75295038303998343806976, 6098898102623865848365056, 524505236825652462959394816

Offset

0,3

Comments

a(n), n>=1, enumerates increasing sextic (6-ary) trees with n vertices. - Wolfdieter Lang, Sep 14 2007

Hankel transform is A169620. - Paul Barry, Dec 03 2009

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..300 (first 50 terms from T. D. Noe)

Martin Burtscher, Igor Szczyrba, Rafał Szczyrba, Analytic Representations of the n-anacci Constants and Generalizations Thereof, Journal of Integer Sequences, Vol. 18 (2015), Article 15.4.5.

Wolfdieter Lang, On generalizations of Stirling number triangles, J. Integer Seqs., Vol. 3 (2000), Article 00.2.4.

J.-C. Novelli and J.-Y. Thibon, Hopf Algebras of m-permutations,(m+1)-ary trees, and m-parking functions, arXiv preprint arXiv:1403.5962 [math.CO], 2014-2020.

Maxie D. Schmidt, Generalized j-Factorial Functions, Polynomials, and Applications , J. Int. Seq. 13 (2010), Article 10.6.7, Table 6.3.

Formula

a(n) = A049385(n, 1) (first column of triangle).

E.g.f.: (1-5*x)^(-1/5).

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

a(n) = Sum_{k=0..n} (-5)^(n-k)*A048994(n, k). - Philippe Deléham, Oct 29 2005

G.f.: 1/(1-x/(1-5x/(1-6x/(1-10x/(1-11x/(1-15x/(1-16x/(1-20x/(1-21x/(1-25x/(1-.../(1-A008851(n+1)*x/(1-... (continued fraction). - Paul Barry, Dec 03 2009

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

G.f.: G(0)/2, where G(k)= 1 + 1/(1 - (5*k+1)*x/((5*k+1)*x + 1/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jun 14 2013

a(n) = (10n-18)*a(n-2) + (5n-6)*a(n-1), n>=2. - Ivan N. Ianakiev, Aug 12 2013

Let T(x) = 1/(1 - 4*x)^(1/4) be the e.g.f. for the sequence of triple factorial numbers A007696. Then the e.g.f. A(x) for the quintuple factorial numbers satisfies T( int {0..x} A(t) dt ) = A(x). Cf. A007559 and A007696. - Peter Bala, Jan 02 2015

O.g.f.: hypergeom([1, 1/5], [], 5*x). - Peter Luschny, Oct 08 2015

a(n) = 5^n * Gamma(n + 1/5) / Gamma(1/5). - Artur Jasinski, Aug 23 2016

D-finite with recurrence: a(n) +(-5*n+4)*a(n-1)=0. - R. J. Mathar, Jan 17 2020

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

Maple

a := n -> mul(5*k+1, k=0..n-1);
G(x):=(1-5*x)^(-1/5): f[0]:=G(x): for n from 1 to 29 do f[n]:=diff(f[n-1], x) od: x:=0: seq(f[n], n=0..16); # Zerinvary Lajos, Apr 03 2009
H := hypergeom([1, 1/5], [], 5*x):
seq(coeff(series(H, x, 20), x, n), n=0..16); # Peter Luschny, Oct 08 2015

Mathematica

Table[Product[5k+1, {k, 0, n-1}], {n, 0, 20}]  (* Harvey P. Dale, Apr 23 2011 *)
FoldList[Times, 1, NestList[#+5&, 1, 20]] (* Ray Chandler, Apr 23 2011 *)
FoldList[Times, 1, 5Range[0, 25] + 1] (* Vincenzo Librandi, Jun 10 2013 *)

Prog

(PARI) x='x+O('x^33); Vec(serlaplace((1-5*x)^(-1/5))) \\ Joerg Arndt, Apr 24 2011
(PARI) vector(20, n, n--; prod(k=0, n-1, 5*k+1)) \\ Altug Alkan, Oct 08 2015
(Magma) [(&*[5*k+1: k in [0..n]]): n in [0..20]]; // G. C. Greubel, Aug 16 2019
(Sage) [product(5*k+1 for k in (0..n)) for n in (0..20)] # G. C. Greubel, Aug 16 2019
(GAP) List([0..20], n-> Product([0..n], k-> 5*k+1)); # G. C. Greubel, Aug 16 2019

Crossrefs

Cf. A001147, A007559, A007696, A016861, A034687, A034688, A052562, A047055, A049385, A051150.

Sequence in context: A360236 A128319 A174496 * A090358 A359462 A264407

Adjacent sequences: A008545 A008546 A008547 * A008549 A008550 A008551

Keyword

nonn,nice,easy

Author

Joe Keane (jgk(AT)jgk.org)

Status

approved