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%I #31 Dec 19 2022 03:45:34
%S 1,9,126,2394,57456,1666224,56651616,2209413024,97214173056,
%T 4763494479744,257228701906176,15176493412464384,971295578397720576,
%U 67019394909442719744,4959435223298761261056,391795382640602139623424,32910812141810579728367616,2929062280621141595824717824
%N a(n) = n-th quintic factorial number divided by 4.
%H G. C. Greubel, <a href="/A034301/b034301.txt">Table of n, a(n) for n = 1..350</a>
%F a(n) = A008546(n)/4.
%F 4*a(n) = (5*n-1)(!^5) = Product_{j=1..n} (5*j-1).
%F a(n) = (5*n)!/(5^n*n!*A008548(n)*2*A034323(n)*3*A034300(n)).
%F E.g.f.: (-1 + (1-5*x)^(-4/5))/4, a(0) = 0.
%F a(n) ~ sqrt(2*Pi) * 5/(4*Gamma(4/5)) * n^(13/10) * (5*n/e)^n * (1 + (241/300)/n + ...). - Joe Keane (jgk(AT)jgk.org), Nov 24 2001
%F D-finite with recurrence: a(n) +(-5*n+1)*a(n-1)=0. - _R. J. Mathar_, Feb 20 2020
%F Sum_{n>=1} 1/a(n) = 4*(e/5)^(1/5)*(Gamma(4/5) - Gamma(4/5, 1/5)). - _Amiram Eldar_, Dec 19 2022
%p a:= n-> mul(5*k-1, k=1..n)/4: seq(a(n), n=1..20); # _G. C. Greubel_, Aug 23 2019
%t Table[-5^(n+1)*Pochhammer[-1/5, n+1]/4, {n,20}] (* _G. C. Greubel_, Aug 23 2019 *)
%o (PARI) a(n) = prod(k=1,n, 5*k-1)/4;
%o vector(20, n, a(n)) \\ _G. C. Greubel_, Aug 23 2019
%o (Magma) [&*[5*k-1: k in [1..n]]/4: n in [1..20]]; // _G. C. Greubel_, Aug 23 2019
%o (Sage) [-5^(n+1)*rising_factorial(-1/5, n+1)/4 for n in (1..20)] # _G. C. Greubel_, Aug 23 2019
%o (GAP) List([1..20], n-> Product([1..n], k-> 5*k-1)/4 ); # _G. C. Greubel_, Aug 23 2019
%Y Cf. A008546, A008548, A034300, A025750.
%K easy,nonn
%O 1,2
%A _Wolfdieter Lang_
%E Terms a(17) onward added by _G. C. Greubel_, Aug 23 2019
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