A169620 Hankel transform of quintuple factorial numbers A008548.

1, 5, 1500, 74250000, 1176120000000000, 9780613920000000000000000, 63441756579801600000000000000000000000, 446492348463430358369280000000000000000000000000000000

Offset: 0

Author

Paul Barry, Dec 03 2009

Keywords

• easy
• nonn

Links

• G. C. Greubel, Table of n, a(n) for n = 0..25

Crossrefs

Cf. A008548.

Programs

GAP

List([0..10], n-> Product([0..n], k-> ((5*k+1)*(5*k+5))^(n-k))); # G. C. Greubel, Aug 17 2019

Magma

[(&*[((5*k+1)*(5*k+5))^(n-k): k in [0..n]]): n in [0..10]]; // G. C. Greubel, Aug 17 2019

Maple

seq(product(((5*k+1)*(5*k+5))^(n-k), k = 0..n), n = 0..10); # G. C. Greubel, Aug 17 2019

Mathematica

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

PARI

vector(10, n, n--; prod(k=0,n, ((5*k+1)*(5*k+5))^(n-k))) \\ G. C. Greubel, Aug 17 2019

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

Formula

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

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