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a(n) = C(2n,n) * (8^n/n!^2) * Product_{k=0..n-1} (8k+1)*(8k+7).
5

%I #10 Oct 05 2020 06:10:25

%S 1,112,90720,105100800,142542960000,211337613527040,

%T 331831362513530880,542307255307827609600,912855634598629193472000,

%U 1571864775032876891607040000,2755743023914838714304931102720

%N a(n) = C(2n,n) * (8^n/n!^2) * Product_{k=0..n-1} (8k+1)*(8k+7).

%F Self-convolution of A184897, where A184897(n) = (8^n/n!^2) * Product_{k=0..n-1} (16k+1)*(16k+7).

%F a(n) ~ sqrt(2-sqrt(2)) * 2^(11*n - 1) / (Pi^(3/2) * n^(3/2)). - _Vaclav Kotesovec_, Oct 05 2020

%e G.f.: A(x) = 1 + 112*x + 90720*x^2 + 105100800*x^3 +...

%e A(x)^(1/2) = 1 + 56*x + 43792*x^2 + 50098048*x^3 +...+ A184897(n)*x^n +...

%o (PARI) {a(n)=(2*n)!/n!^2*(8^n/n!^2)*prod(k=0,n-1,(8*k+1)*(8*k+7))}

%Y Cf. A184897; variants: A184423, A008977, A184892, A001421, A184896.

%K nonn

%O 0,2

%A _Paul D. Hanna_, Jan 25 2011