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A034787 a(n) = n-th sextic factorial number divided by 5.

%I A034787 #32 Apr 04 2025 07:18:06
%S A034787 1,11,187,4301,124729,4365515,178986115,8412347405,445854412465,
%T A034787 26305410335435,1709851671803275,121399468698032525,
%U A034787 9347759089748504425,775864004449125867275,69051896395972202187475,6559930157617359207810125,662552945919353279988822625
%N A034787 a(n) = n-th sextic factorial number divided by 5.
%H A034787 G. C. Greubel, <a href="/A034787/b034787.txt">Table of n, a(n) for n = 1..345</a>
%F A034787 5*a(n) = (6*n-1)(!^6) = Product_{j=1..n} (6*j-1) = (6*n)!/(3^(2*n)*2^(2*n+1)*(2*n)!*A008542(n)*A007559(n)*A034000(n)).
%F A034787 E.g.f.: (-1 + (1-6*x)^(-5/6))/5.
%F A034787 a(n+1) ~ sqrt(2*Pi) * 6/(5*Gamma(5/6)) * n^(4/3) * (6*n/e)^n * (1 + (61/72)/n + ...). - Joe Keane (jgk(AT)jgk.org), Nov 24 2001
%F A034787 D-finite with recurrence: a(n) +(-6*n+1)*a(n-1)=0. - _R. J. Mathar_, Feb 24 2020
%F A034787 Sum_{n>=1} 1/a(n) = 5*(e/6)^(1/6)*(Gamma(5/6) - Gamma(5/6, 1/6)). - _Amiram Eldar_, Dec 18 2022
%p A034787 seq( mul(6*j-1, j=1..n)/5, n=1..20); # _G. C. Greubel_, Nov 11 2019
%t A034787 Table[6^n*Pochhammer[5/6, n]/5, {n,20}] (* _G. C. Greubel_, Nov 11 2019 *)
%t A034787 With[{nn=20},CoefficientList[Series[(-1+(1-6x)^(-5/6))/5,{x,0,nn}],x] Range[0,nn]!] (* _Harvey P. Dale_, Dec 21 2024 *)
%o A034787 (PARI) vector(20, n, prod(j=1,n, 6*j-1)/5 ) \\ _G. C. Greubel_, Nov 11 2019
%o A034787 (Magma) [(&*[6*j-1: j in [1..n]])/5: n in [1..20]]; // _G. C. Greubel_, Nov 11 2019
%o A034787 (Sage) [product( (6*j-1) for j in (1..n))/5 for n in (1..20)] # _G. C. Greubel_, Nov 11 2019
%o A034787 (GAP) List([1..20], n-> Product([1..n], j-> 6*j-1)/5 ); # _G. C. Greubel_, Nov 11 2019
%Y A034787 Cf. A008542, A034689, A034723, A034724, A034788.
%K A034787 easy,nonn
%O A034787 1,2
%A A034787 _Wolfdieter Lang_