%I #9 Jan 23 2024 08:27:09

%S 1,5,1500,74250000,1176120000000000,9780613920000000000000000,

%T 63441756579801600000000000000000000000,

%U 446492348463430358369280000000000000000000000000000000

%N Hankel transform of quintuple factorial numbers A008548.

%H G. C. Greubel, <a href="/A169620/b169620.txt">Table of n, a(n) for n = 0..25</a>

%F a(n) = Product_{k=0..n} ((5*k+1)*(5*k+5))^(n-k).

%F a(n) ~ (2*Pi)^(n + 3/5) * 5^(n*(n+1)) * n^(n^2 + 6*n/5 + 53/150) / (A * Gamma(1/5)^(n + 1/5) * exp(3*n^2/2 + 6*n/5 - 1/12 - c)), where A is the Glaisher-Kinkelin constant A074962 and c = zeta'(-1, 1/5) = 0.0831827651866002925663000008102352492418540625037508868... - _Vaclav Kotesovec_, Jan 23 2024

%p seq(product(((5*k+1)*(5*k+5))^(n-k), k = 0..n), n = 0..10); # _G. C. Greubel_, Aug 17 2019

%t Table[Product[((5*k+1)*(5*k+5))^(n-k), {k,0,n}], {n,0,10}] (* _G. C. Greubel_, Aug 17 2019 *)

%o (PARI) vector(10, n, n--; prod(k=0,n, ((5*k+1)*(5*k+5))^(n-k))) \\ _G. C. Greubel_, Aug 17 2019

%o (Magma) [(&*[((5*k+1)*(5*k+5))^(n-k): k in [0..n]]): n in [0..10]]; // _G. C. Greubel_, Aug 17 2019

%o (Sage) [product(((5*k+1)*(5*k+5))^(n-k) for k in (0..n)) for n in (0..10)] # _G. C. Greubel_, Aug 17 2019

%o (GAP) List([0..10], n-> Product([0..n], k-> ((5*k+1)*(5*k+5))^(n-k))); # _G. C. Greubel_, Aug 17 2019

%Y Cf. A008548.

%K easy,nonn

%O 0,2

%A _Paul Barry_, Dec 03 2009