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One quarter of deca-factorial numbers.
10

%I #20 Dec 22 2022 07:22:28

%S 1,14,336,11424,502656,27143424,1737179136,128551256064,

%T 10798305509376,1015040717881344,105564234659659776,

%U 12034322751201214464,1492256021148950593536,199962306833959379533824,28794572184090150652870656,4434364116349883200542081024

%N One quarter of deca-factorial numbers.

%H G. C. Greubel, <a href="/A035273/b035273.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 4*a(n) = (10*n-6)(!^10) = Product_{j=1..n} (10*j-6).

%F a(n) = 2^(n+1)*A034323(n) where 2*A034323(n)= (5*n-3)(!^5) = Product_{j=1..n} (5*j-3).

%F E.g.f.: (-1 + (1-10*x)^(-2/5))/4.

%F a(n) = (Pochhammer(4/10,n)*10^n)/4.

%F Sum_{n>=1} 1/a(n) = 4*(e/10^6)^(1/10)*(Gamma(2/5) - Gamma(2/5, 1/10)). - _Amiram Eldar_, Dec 22 2022

%p seq( mul(10*j-6, j=1..n)/4, n=1..20); # _G. C. Greubel_, Nov 11 2019

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

%o (PARI) vector(20, n, prod(j=1,n, 10*j-6)/4 ) \\ _G. C. Greubel_, Nov 11 2019

%o (Magma) [(&*[10*j-6: j in [1..n]])/4: n in [1..20]]; // _G. C. Greubel_, Nov 11 2019

%o (Sage) [product( (10*j-6) for j in (1..n))/4 for n in (1..20)] # _G. C. Greubel_, Nov 11 2019

%o (GAP) List([1..20], n-> Product([1..n], j-> 10*j-6)/4 ); # _G. C. Greubel_, Nov 11 2019

%Y Cf. A034323, A035272, A035274, A035275, A035276, A035277, A035278, A035279, A045757.

%K easy,nonn

%O 1,2

%A _Wolfdieter Lang_