A184892 a(n) = C(2n,n) * (5^n/n!^2) * Product_{k=0..n-1} (5k+1)*(5k+4).
1, 40, 8100, 2310000, 768075000, 278719056000, 107022956040000, 42753018765600000, 17585519046944062500, 7397979398239787500000, 3168258657090171394750000, 1376657183877933677265000000
Offset: 0
Keywords
Examples
G.f.: A(x) = 1 + 40*x + 8100*x^2 + 2310000*x^3 +... A(x)^(1/2) = 1 + 20*x + 3850*x^2 + 1078000*x^3 +...+ A184891(n)*x^n +...
Programs
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Mathematica
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 *) -
PARI
{a(n)=(2*n)!/n!^2*(5^n/n!^2)*prod(k=0,n-1,(5*k+1)*(5*k+4))}
Formula
Self-convolution of A184891, where
. A184891(n) = (5^n/n!^2) * Product_{k=0..n-1} (10k+1)*(10k+4).
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