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One eighth of deca-factorial numbers.
10

%I #18 Dec 22 2022 04:15:46

%S 1,18,504,19152,919296,53319168,3625703424,282804867072,

%T 24886828302336,2438909173628928,263402190751924224,

%U 31081458508727058432,3978426689117063479296,549022883098154760142848,81255386698526904501141504,12838351098367250911180357632

%N One eighth of deca-factorial numbers.

%H G. C. Greubel, <a href="/A035277/b035277.txt">Table of n, a(n) for n = 1..320</a>

%H <a href="/index/Fa#factorial">Index entries for sequences related to factorial numbers</a>.

%F a(n) = (Pochhammer(8/10,n)*10^n)/8.

%F 8*a(n) = (10*n-2)(!^10) = Product_{j=1..n} (10*j-2).

%F a(n) = 2^(n+2)*A034301(n) where 4*A034301(n) = (5*n-1)(!^5).

%F E.g.f.: (-1 + (1-10*x)^(-4/5))/8.

%F Sum_{n>=1} 1/a(n) = 8*(e/10^2)^(1/10)*(Gamma(4/5) - Gamma(4/5, 1/10)). - _Amiram Eldar_, Dec 22 2022

%p seq( mul(10*j-2, j=1..n)/8, n=1..20); # _G. C. Greubel_, Nov 11 2019

%t Table[10^n*Pochhammer[4/5, n]/8, {n,20}] (* _G. C. Greubel_, Nov 11 2019 *)

%o (PARI) vector(20, n, prod(j=1,n, 10*j-2)/8 ) \\ _G. C. Greubel_, Nov 11 2019

%o (Magma) [(&*[10*j-2: j in [1..n]])/8: n in [1..20]]; // _G. C. Greubel_, Nov 11 2019

%o (Sage) [product( (10*j-2) for j in (1..n))/8 for n in (1..20)] # _G. C. Greubel_, Nov 11 2019

%o (GAP) List([1..20], n-> Product([1..n], j-> 10*j-2)/8 ); # _G. C. Greubel_, Nov 11 2019

%Y Cf. A034301, A045757, A035265, A035272, A035273, A035274, A035275, A035276, A035277, A035278, A035279.

%K easy,nonn

%O 1,2

%A _Wolfdieter Lang_