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A185403 a(n) = (7^n/n!^2) * Product_{k=0..n-1} (14k+3)*(14k+4).

%I A185403 #11 Nov 19 2023 07:28:35
%S A185403 1,84,44982,34706112,31430722680,31154132320416,32723954432339184,
%T A185403 35790656447712684672,40328240610474258475572,
%U A185403 46491988990198595758628560,54576945875594131561054066584
%N A185403 a(n) = (7^n/n!^2) * Product_{k=0..n-1} (14k+3)*(14k+4).
%H A185403 G. C. Greubel, <a href="/A185403/b185403.txt">Table of n, a(n) for n = 0..320</a>
%F A185403 Self-convolution yields Sum_{k=0..n} a(n-k)*a(k) = A185404(n) where A185404(n) = C(2n,n) * (7^n/n!^2)*Product_{k=0..n-1} (7k+3)*(7k+4).
%F A185403 a(n) ~ 2^(2*n) * 7^(3*n) / (Gamma(2/7) * Gamma(3/14) * n^(3/2)). - _Vaclav Kotesovec_, Nov 19 2023
%e A185403 G.f.: A(x) = 1 + 84*x + 44982*x^2 + 34706112*x^3 +...
%e A185403 A(x)^2 = 1 + 168*x + 97020*x^2 + 76969200*x^3 +...+ A185404(n)*x^n +...
%t A185403 Table[(7^n/(n!)^2)*Product[(14*k + 3)*(14*k + 4), {k, 0, n - 1}], {n, 0, 50}] (* _G. C. Greubel_, Jun 29 2017 *)
%o A185403 (PARI) {a(n)=(7^n/n!^2)*prod(k=0,n-1,(14*k+3)*(14*k+4))}
%Y A185403 Cf. A184895, A185401, A185404.
%K A185403 nonn
%O A185403 0,2
%A A185403 _Paul D. Hanna_, Jan 26 2011