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A034300 a(n) = n-th quintic factorial number divided by 3.

%I A034300 #41 Dec 19 2022 03:45:45
%S A034300 1,8,104,1872,43056,1205568,39783744,1511782272,65006637696,
%T A034300 3120318609408,165376886298624,9591859405320192,604287142535172096,
%U A034300 41091525692391702528,2999681375544594284544,233975147292478354194432,19419937225275703398137856,1708954475824261899036131328
%N A034300 a(n) = n-th quintic factorial number divided by 3.
%H A034300 G. C. Greubel, <a href="/A034300/b034300.txt">Table of n, a(n) for n = 1..350</a>
%F A034300 3*a(n) = (5*n-2)(!^5) = Product_{j=1..n} (5*j-2) = A047056(n).
%F A034300 E.g.f.: (1-5*x)^(-3/5)/3.
%F A034300 a(n) ~ sqrt(2*Pi) * 5/(3*Gamma(3/5)) * n^(11/10) * (5*n/e)^n * (1 + (169/300)/n - ...). - Joe Keane (jgk(AT)jgk.org), Nov 24 2001
%F A034300 a(n) = 5^n * Pochhammer(3/5, n)/3. - _G. C. Greubel_, Aug 23 2019
%F A034300 D-finite with recurrence: a(n) +(-5*n+2)*a(n-1)=0. - _R. J. Mathar_, Feb 20 2020
%F A034300 Sum_{n>=1} 1/a(n) = 3*(e/5^2)^(1/5)*(Gamma(3/5) - Gamma(3/5, 1/5)). - _Amiram Eldar_, Dec 19 2022
%p A034300 a:= n-> mul(5*k-2, k=1..n)/3; seq(a(n), n=1..25); # _G. C. Greubel_, Aug 23 2019
%t A034300 Table[Product[5j-2,{j,n}],{n,20}]*1/3 (* _Harvey P. Dale_, Jul 25 2011 *)
%o A034300 (PARI) a(n) = prod(k=1,n, 5*k-2)/3;
%o A034300 vector(25, n, a(n)) \\ _G. C. Greubel_, Aug 23 2019
%o A034300 (Magma) [(&*[5*k-2: k in [1..n]])/3: n in [1..25]]; // _G. C. Greubel_, Aug 23 2019
%o A034300 (Sage) [5^n*rising_factorial(3/5, n)/3 for n in (1..25)] # _G. C. Greubel_, Aug 23 2019
%o A034300 (GAP) List([1..25], n-> Product([1..n], k-> 5*k-2)/3 ); # _G. C. Greubel_, Aug 23 2019
%Y A034300 Cf. A008548, A034301, A047056.
%K A034300 nonn
%O A034300 1,2
%A A034300 _Wolfdieter Lang_
%E A034300 Terms a(17) onward added by _G. C. Greubel_, Aug 23 2019