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

%I A008548 #105 Mar 07 2025 09:36:50
%S A008548 1,1,6,66,1056,22176,576576,17873856,643458816,26381811456,
%T A008548 1213563326976,61891729675776,3465936861843456,211422148572450816,
%U A008548 13953861805781753856,990724188210504523776,75295038303998343806976,6098898102623865848365056,524505236825652462959394816
%N A008548 Quintuple factorial numbers: Product_{k=0..n-1} (5*k+1).
%C A008548 a(n), n>=1, enumerates increasing sextic (6-ary) trees with n vertices. - _Wolfdieter Lang_, Sep 14 2007
%C A008548 Hankel transform is A169620. - _Paul Barry_, Dec 03 2009
%H A008548 Vincenzo Librandi, <a href="/A008548/b008548.txt">Table of n, a(n) for n = 0..300</a> (first 50 terms from T. D. Noe)
%H A008548 Martin Burtscher, Igor Szczyrba, Rafał Szczyrba, <a href="http://www.emis.de/journals/JIS/VOL18/Szczyrba/sz3.html">Analytic Representations of the n-anacci Constants and Generalizations Thereof</a>, Journal of Integer Sequences, Vol. 18 (2015), Article 15.4.5.
%H A008548 Wolfdieter Lang, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL3/LANG/lang.html">On generalizations of Stirling number triangles</a>, J. Integer Seqs., Vol. 3 (2000), Article 00.2.4.
%H A008548 J.-C. Novelli and J.-Y. Thibon, <a href="http://arxiv.org/abs/1403.5962">Hopf Algebras of m-permutations,(m+1)-ary trees, and m-parking functions</a>, arXiv preprint arXiv:1403.5962 [math.CO], 2014-2020.
%H A008548 Maxie D. Schmidt, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL13/Schmidt/multifact.html">Generalized j-Factorial Functions, Polynomials, and Applications </a>, J. Int. Seq. 13 (2010), Article 10.6.7, Table 6.3.
%F A008548 a(n) = A049385(n, 1) (first column of triangle).
%F A008548 E.g.f.: (1-5*x)^(-1/5).
%F A008548 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
%F A008548 a(n) = Sum_{k=0..n} (-5)^(n-k)*A048994(n, k). - _Philippe Deléham_, Oct 29 2005
%F A008548 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
%F A008548 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
%F A008548 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
%F A008548 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
%F A008548 a(n) = (10n-18)*a(n-2) + (5n-6)*a(n-1), n>=2. - _Ivan N. Ianakiev_, Aug 12 2013
%F A008548 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( Integral_{t = 0..x} A(t) dt ) = A(x). Cf. A007559 and A007696. - _Peter Bala_, Jan 02 2015
%F A008548 O.g.f.: hypergeom([1, 1/5], [], 5*x). - _Peter Luschny_, Oct 08 2015
%F A008548 a(n) = 5^n * Gamma(n + 1/5) / Gamma(1/5). - _Artur Jasinski_, Aug 23 2016
%F A008548 D-finite with recurrence: a(n) +(-5*n+4)*a(n-1)=0. - _R. J. Mathar_, Jan 17 2020
%F A008548 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
%p A008548 a := n -> mul(5*k+1, k=0..n-1);
%p A008548 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
%p A008548 H := hypergeom([1, 1/5], [], 5*x):
%p A008548 seq(coeff(series(H,x,20),x,n),n=0..16); # _Peter Luschny_, Oct 08 2015
%t A008548 Table[Product[5k+1,{k,0,n-1}],{n,0,20}]  (* _Harvey P. Dale_, Apr 23 2011 *)
%t A008548 FoldList[Times,1,NestList[#+5&,1,20]] (* _Ray Chandler_, Apr 23 2011 *)
%t A008548 FoldList[Times,1,5Range[0, 25] + 1] (* _Vincenzo Librandi_, Jun 10 2013 *)
%o A008548 (PARI) x='x+O('x^33); Vec(serlaplace((1-5*x)^(-1/5))) \\ _Joerg Arndt_, Apr 24 2011
%o A008548 (PARI) vector(20, n, n--; prod(k=0, n-1, 5*k+1)) \\ _Altug Alkan_, Oct 08 2015
%o A008548 (Magma) [(&*[5*k+1: k in [0..n]]): n in [0..20]]; // _G. C. Greubel_, Aug 16 2019
%o A008548 (Sage) [product(5*k+1 for k in (0..n)) for n in (0..20)] # _G. C. Greubel_, Aug 16 2019
%o A008548 (GAP) List([0..20], n-> Product([0..n], k-> 5*k+1)); # _G. C. Greubel_, Aug 16 2019
%Y A008548 Cf. A001147, A007559, A007696, A016861, A034687, A034688, A052562, A047055, A049385, A051150.
%K A008548 nonn,nice,easy
%O A008548 0,3
%A A008548 Joe Keane (jgk(AT)jgk.org)