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 A184890 a(n) = C(2n,n) * (5^n/n!^2) * Product_{k=0..n-1} (5k+2)*(5k+3). 2

%I #7 Oct 07 2020 07:47:47

%S 1,60,12600,3640000,1218262500,443837394000,170877396690000,

%T 68390813462400000,28171137810976875000,11864338450927462500000,

%U 5085530033605547526000000,2211345876971860770960000000

%N a(n) = C(2n,n) * (5^n/n!^2) * Product_{k=0..n-1} (5k+2)*(5k+3).

%F Self-convolution of A184889:

%F A184889(n) = (5^n/n!^2) * Product_{k=0..n-1} (10k+2)*(10k+3).

%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 + 60*x + 12600*x^2 + 3640000*x^3 +...

%e A(x)^(1/2) = 1 + 30*x + 5850*x^2 + 1644500*x^3 +...+ A184889(n)*x^n +...

%t Table[Binomial[2*n, n] * 5^n / n!^2 * Product[(5*k + 2)*(5*k + 3), {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+2)*(5*k+3))}

%Y Cf. A184889, A184892.

%K nonn

%O 0,2

%A _Paul D. Hanna_, Jan 25 2011

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Last modified July 17 11:42 EDT 2024. Contains 374377 sequences. (Running on oeis4.)