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A008542 Sextuple factorial numbers: Product_{k=0..n-1} (6*k+1). 39

%I #64 Dec 18 2022 02:25:02

%S 1,1,7,91,1729,43225,1339975,49579075,2131900225,104463111025,

%T 5745471106375,350473737488875,23481740411754625,1714167050058087625,

%U 135419196954588922375,11510631741140058401875,1047467488443745314570625,101604346379043295513350625

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

%C a(n), n>=1, enumerates increasing heptic (7-ary) trees with n vertices. - _Wolfdieter Lang_, Sep 14 2007; see a D. Callan comment on A007559 (number of increasing quarterny trees).

%H Vincenzo Librandi, <a href="/A008542/b008542.txt">Table of n, a(n) for n = 0..300</a>

%H <a href="/index/Fa#factorial">Index entries for sequences related to factorial numbers</a>.

%F E.g.f.: (1-6*x)^(-1/6).

%F a(n) ~ 2^(1/2)*Pi^(1/2)*Gamma(1/6)^-1*n^(-1/3)*6^n*e^-n*n^n*{1 + 1/72*n^-1 - ...}. - Joe Keane (jgk(AT)jgk.org), Nov 24 2001

%F a(n) = Sum_{k=0..n} (-6)^(n-k)*A048994(n, k). - _Philippe Deléham_, Oct 29 2005

%F G.f.: 1+x/(1-7x/(1-6x/(1-13x/(1-12x/(1-19x/(1-18x/(1-25x/(1-24x/(1-... (continued fraction). - _Philippe Deléham_, Jan 08 2012

%F a(n) = (-5)^n*Sum_{k=0..n} (6/5)^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

%F G.f.: 1/Q(0) where Q(k) = 1 - x*(6*k+1)/(1 - x*(6*k+6)/Q(k+1) ); (continued fraction). - _Sergei N. Gladkovskii_, Mar 20 2013

%F a(n) = A085158(6*n-5). - _M. F. Hasler_, Feb 23 2018

%F D-finite with recurrence: a(n) +(-6*n+5)*a(n-1)=0. - _R. J. Mathar_, Jan 17 2020

%F Sum_{n>=0} 1/a(n) = 1 + (e/6^5)^(1/6)*(Gamma(1/6) - Gamma(1/6, 1/6)). - _Amiram Eldar_, Dec 18 2022

%p a := n -> mul(6*k+1, k=0..n-1);

%p G(x):=(1-6*x)^(-1/6): 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..15); # _Zerinvary Lajos_, Apr 03 2009

%t Table[Product[(6*k+1), {k,0,n-1}], {n,0,20}] (* _Vladimir Joseph Stephan Orlovsky_, Nov 08 2008, modified by _G. C. Greubel_, Aug 17 2019 *)

%t FoldList[Times, 1, 6Range[0, 20] + 1] (* _Vincenzo Librandi_, Jun 10 2013 *)

%t Table[6^n*Pochhammer[1/6, n], {n,0,20}] (* _G. C. Greubel_, Aug 17 2019 *)

%o (PARI) a(n)=prod(k=1,n-1,6*k+1) \\ _Charles R Greathouse IV_, Jul 19 2011

%o (Magma) [1] cat [(&*[(6*k+1): k in [0..n-1]]): n in [1..20]]; // _G. C. Greubel_, Aug 17 2019

%o (Sage) [product((6*k+1) for k in (0..n-1)) for n in (0..20)] # _G. C. Greubel_, Aug 17 2019

%o (GAP) List([0..20], n-> Product([0..n-1], k-> (6*k+1) )); # _G. C. Greubel_, Aug 17 2019

%Y Cf. A085158, A034689, A034723, A034724, A034787, A034788, A004993, A047058, A047657, A051151.

%Y Cf. k-fold factorials: A000142, A001147 (and A000165, A006882), A007559 (and A032031, A008544, A007661), A007696 (and A001813, A008545, A047053, A007662), A008548 (and A052562, A047055, A085157), A045754 (and A084947, A114799), A045755.

%K nonn

%O 0,3

%A Joe Keane (jgk(AT)jgk.org)

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Last modified March 29 01:36 EDT 2024. Contains 371264 sequences. (Running on oeis4.)