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%I A034323 #48 Dec 19 2022 03:45:00
%S A034323 1,7,84,1428,31416,848232,27143424,1004306688,42180880896,
%T A034323 1982501402112,103090072909824,5876134155859968,364320317663318016,
%U A034323 24409461283442307072,1757481212407846109184,135326053355404150407168,11096736375143140333387776,965416064637453209004736512
%N A034323 a(n) = n-th quintic factorial number divided by 2.
%H A034323 Robert Israel, <a href="/A034323/b034323.txt">Table of n, a(n) for n = 1..355</a>
%H A034323 J.-C. Novelli and J.-Y. Thibon, <a href="http://arxiv.org/abs/1403.5962">Hopf Algebras of m-permutations,(m+1)-ary trees, and m-parking functions</a>, arXiv:1403.5962 [math.CO], 2014-2020.
%F A034323 2*a(n) = (5*n-3)(!^5) = Product_{j=1..n} (5*j-3).
%F A034323 E.g.f.: (-1 + (1-5*x)^(-2/5))/2, with a(0) = 0.
%F A034323 a(n) ~ sqrt(2*Pi) * 5/(2*Gamma(2/5)) * n^(9/10) * (5*n/e)^n * (1 + (109/300)/n - ...). - Joe Keane (jgk(AT)jgk.org), Nov 24 2001
%F A034323 D-finite with recurrence a(n) = (5*n-3)*a(n-1). - _Robert Israel_, Feb 10 2019
%F A034323 From _Amiram Eldar_, Dec 19 2022: (Start)
%F A034323 a(n) = A047055(n)/2.
%F A034323 Sum_{n>=1} 1/a(n) = 2*(e/5^3)^(1/5)*(Gamma(2/5) - Gamma(2/5, 1/5)). (End)
%p A034323 f:= gfun:-rectoproc({a(n)=(5*n-3)*a(n-1),a(1)=1},a(n),remember):
%p A034323 map(f, [$1..40]); # _Robert Israel_, Feb 10 2019
%t A034323 Table[Product[5j-3,{j,n}]/2,{n,20}] (* _Harvey P. Dale_, Nov 25 2013 *)
%o A034323 (PARI) vector(20, n, prod(j=1, n, 5*j-3)/2) \\ _G. C. Greubel_, Feb 10 2019
%o A034323 (Magma) [(&*[5*j-3: j in [1..n]])/2: n in [1..20]]; // _G. C. Greubel_, Feb 10 2019
%o A034323 (Sage) [product(5*j-3 for j in (1..n))/2 for n in (1..20)] # _G. C. Greubel_, Feb 10 2019
%o A034323 (GAP) a:=[1];; for n in [2..20] do a[n]:=(5*n-3)*a[n-1]; od; a; # _G. C. Greubel_, Feb 10 2019
%Y A034323 Cf. A008548, A034300, A034301, A034325, A047055.
%K A034323 easy,nonn
%O A034323 1,2
%A A034323 _Wolfdieter Lang_

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