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A184895 a(n) = (7^n/n!^2) * Product_{k=0..n-1} (14k+1)*(14k+6).

%I A184895 #15 Nov 19 2023 07:32:34
%S A184895 1,42,22050,16909900,15269639700,15109613875944,15853342647837688,
%T A184895 17325438750851187600,19510609713302293636050,
%U A184895 22482485054570487449402900,26382746561837375612125315092,31419888802098260334367621118904
%N A184895 a(n) = (7^n/n!^2) * Product_{k=0..n-1} (14k+1)*(14k+6).
%H A184895 Vincenzo Librandi, <a href="/A184895/b184895.txt">Table of n, a(n) for n = 0..100</a>
%F A184895 Self-convolution yields Sum_{k=0..n} a(n-k)*a(k) = A184896(n) where A184896(n) = C(2n,n) * (7^n/n!^2)*Product_{k=0..n-1} (7k+1)*(7k+6).
%F A184895 a(n) ~ 2^(2*n) * 7^(3*n) / (Gamma(3/7) * Gamma(1/14) * n^(3/2)). - _Vaclav Kotesovec_, Nov 19 2023
%e A184895 G.f.: A(x) = 1 + 42*x + 22050*x^2 + 16909900*x^3 +...
%e A184895 A(x)^2 = 1 + 84*x + 45864*x^2 + 35672000*x^3 +...+ A184896(n)*x^n +...
%t A184895 FullSimplify[Table[2^(2*n) * 7^(3*n) * Gamma[n+1/14] * Gamma[n+3/7] / (Gamma[3/7] * Gamma[1/14] * Gamma[n+1]^2), {n, 0, 15}]] (* _Vaclav Kotesovec_, Jul 03 2014 *)
%o A184895 (PARI) {a(n)=(7^n/n!^2)*prod(k=0,n-1,(14*k+1)*(14*k+6))}
%Y A184895 Cf. A184896; variants: A184424, A178529, A184891, A092870, A184897.
%K A184895 nonn
%O A184895 0,2
%A A184895 _Paul D. Hanna_, Jan 25 2011