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A034325 a(n) is the n-th quintic factorial number divided by 5.

%I A034325 #39 Sep 08 2022 08:44:51
%S A034325 1,10,150,3000,75000,2250000,78750000,3150000000,141750000000,
%T A034325 7087500000000,389812500000000,23388750000000000,1520268750000000000,
%U A034325 106418812500000000000,7981410937500000000000,638512875000000000000000
%N A034325 a(n) is the n-th quintic factorial number divided by 5.
%H A034325 Michael De Vlieger, <a href="/A034325/b034325.txt">Table of n, a(n) for n = 1..354</a>
%H A034325 INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=594">Encyclopedia of Combinatorial Structures 594</a>.
%H A034325 Norihiro Nakashima and Shuhei Tsujie, <a href="https://arxiv.org/abs/1904.09748">Enumeration of Flats of the Extended Catalan and Shi Arrangements with Species</a>, arXiv:1904.09748 [math.CO], 2019.
%F A034325 5*a(n) = (5*n)(!^5) = Product_{j=1..n} 5*j = 5^(n-1)*n!.
%F A034325 E.g.f.: (-1 + (1-5*x)^(-1))/5, a(0) = 0.
%F A034325 D-finite with recurrence: a(n) - 5*n*a(n-1) = 0. - _R. J. Mathar_, Feb 24 2020
%F A034325 From _Amiram Eldar_, Jan 08 2022: (Start)
%F A034325 Sum_{n>=1} 1/a(n) = 5*(exp(1/5)-1).
%F A034325 Sum_{n>=1} (-1)^(n+1)/a(n) = 5*(1-exp(-1/5)). (End)
%p A034325 seq(5^(n-1)*n!, n=1..20); # _G. C. Greubel_, Aug 23 2019
%t A034325 Array[5^(# - 1) #! &, 16] (* _Michael De Vlieger_, May 30 2019 *)
%o A034325 (PARI) vector(20, n, 5^(n-1)*n!) \\ _G. C. Greubel_, Aug 23 2019
%o A034325 (Magma) [5^(n-1)*Factorial(n): n in [1..20]]; // _G. C. Greubel_, Aug 23 2019
%o A034325 (Sage) [5^(n-1)*factorial(n) for n in (1..20)] # _G. C. Greubel_, Aug 23 2019
%o A034325 (GAP) List([1..20], n-> 5^(n-1)*Factorial(n) ); # _G. C. Greubel_, Aug 23 2019
%Y A034325 Cf. A008548, A034300, A034301, A034323, A052562.
%K A034325 easy,nonn
%O A034325 1,2
%A A034325 _Wolfdieter Lang_