login

Reminder: The OEIS is hiring a new managing editor, and the application deadline is January 26.

a(n) = C(2n,n) * (5^n/n!^2) * Product_{k=0..n-1} (5k+1)*(5k+4).
5

%I #10 Oct 07 2020 07:52:54

%S 1,40,8100,2310000,768075000,278719056000,107022956040000,

%T 42753018765600000,17585519046944062500,7397979398239787500000,

%U 3168258657090171394750000,1376657183877933677265000000

%N a(n) = C(2n,n) * (5^n/n!^2) * Product_{k=0..n-1} (5k+1)*(5k+4).

%F Self-convolution of A184891, where

%F . A184891(n) = (5^n/n!^2) * Product_{k=0..n-1} (10k+1)*(10k+4).

%F a(n) ~ sqrt(5 - sqrt(5)) * 2^(2*n - 3/2) * 5^(3*n) / (Pi^(3/2) * n^(3/2)). - _Vaclav Kotesovec_, Oct 07 2020

%e G.f.: A(x) = 1 + 40*x + 8100*x^2 + 2310000*x^3 +...

%e A(x)^(1/2) = 1 + 20*x + 3850*x^2 + 1078000*x^3 +...+ A184891(n)*x^n +...

%t Table[Binomial[2*n, n] * 5^n / n!^2 * Product[(5*k + 1)*(5*k + 4), {k, 0, n - 1}], {n, 0, 20}] (* _Vaclav Kotesovec_, Oct 07 2020 *)

%o (PARI) {a(n)=(2*n)!/n!^2*(5^n/n!^2)*prod(k=0,n-1,(5*k+1)*(5*k+4))}

%Y Cf. A184891, A184890, A184423, A008977, A001421, A184896, A184898.

%K nonn

%O 0,2

%A _Paul D. Hanna_, Jan 25 2011