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 A008548 Quintuple factorial numbers: Product_{k=0..n-1} (5*k+1). 53
 1, 1, 6, 66, 1056, 22176, 576576, 17873856, 643458816, 26381811456, 1213563326976, 61891729675776, 3465936861843456, 211422148572450816, 13953861805781753856, 990724188210504523776, 75295038303998343806976 (list; graph; refs; listen; history; text; internal format)
 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 T. D. Noe and 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. W. Lang, On generalizations of Stirling number triangles, J. Integer Seqs., Vol. 3 (2000), #00.2.4. J.-C. Novelli, J.-Y. Thibon, Hopf Algebras of m-permutations,(m+1)-ary trees, and m-parking functions, arXiv preprint arXiv:1403.5962 [math.CO], 2014. M. D. Schmidt, Generalized j-Factorial Functions, Polynomials, and Applications , J. Int. Seq. 13 (2010), 10.6.7, Table 6.3 FORMULA 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 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, A051150. a(n)= A049385(n, 1) (first column of triangle). Sequence in context: A211824 A128319 A174496 * A090358 A264407 A112942 Adjacent sequences:  A008545 A008546 A008547 * A008549 A008550 A008551 KEYWORD nonn,nice,easy AUTHOR Joe Keane (jgk(AT)jgk.org) STATUS approved

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Last modified August 3 11:49 EDT 2020. Contains 336198 sequences. (Running on oeis4.)