cp's OEIS Frontend

A185404 a(n) = C(2n,n) * (7^n/n!^2) * Product_{k=0..n-1} (7k+3)*(7k+4).

%I A185404 #11 Oct 23 2020 06:41:41
%S A185404 1,168,97020,76969200,70715452500,70710926711040,74713950839848320,
%T A185404 82063363963278297600,92763657280052631873000,
%U A185404 107208829261440251585240000,126104599836427618807641720480
%N A185404 a(n) = C(2n,n) * (7^n/n!^2) * Product_{k=0..n-1} (7k+3)*(7k+4).
%H A185404 G. C. Greubel, <a href="/A185404/b185404.txt">Table of n, a(n) for n = 0..320</a>
%F A185404 Self-convolution of A185403:
%F A185404 A185403(n) = (7^n/n!^2) * Product_{k=0..n-1} (14k+3)*(14k+4).
%F A185404 a(n) ~ cos(Pi/14) * 2^(2*n) * 7^(3*n) / (Pi*n)^(3/2). - _Vaclav Kotesovec_, Oct 23 2020
%e A185404 G.f.: A(x) = 1 + 168*x + 97020*x^2 + 76969200*x^3 +...
%e A185404 A(x)^(1/2) = 1 + 84*x + 44982*x^2 + 34706112*x^3 +...+ A185403(n)*x^n +...
%t A185404 Table[Binomial[2*n, n]*(7^n/(n!)^2)*Product[(7*k + 3)*(7*k + 4), {k, 0, n - 1}], {n, 0, 50}] (* _G. C. Greubel_, Jun 29 2017 *)
%o A185404 (PARI) {a(n)=(2*n)!/n!^2*(7^n/n!^2)*prod(k=0,n-1,(7*k+3)*(7*k+4))}
%Y A185404 Cf. A184896, A185402, A185403.
%K A185404 nonn
%O A185404 0,2
%A A185404 _Paul D. Hanna_, Jan 26 2011