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A383965 Self-convolution square-root of A004381, where A004381(n) = binomial(8*n,n).

%I A383965 #53 Aug 16 2025 10:02:23
%S A383965 1,4,52,804,13412,233548,4180932,76307228,1412731844,26443784224,
%T A383965 499310856828,9494966722696,181620437132820,3491268491768400,
%U A383965 67396227598309788,1305787014634864584,25380012805871145604,494684878753394992992,9665968233663380580256,189289570996914582016788
%N A383965 Self-convolution square-root of A004381, where A004381(n) = binomial(8*n,n).
%C A383965 In general, for m > 1, if Sum_{k=0..n} a(k)*a(n-k) = binomial(m*n,n), then a(n) ~ m^(m*n + 1/4) / (2^(1/4) * Gamma(1/4) * (m-1)^((m-1)*n + 1/4) * n^(3/4)).
%F A383965 a(n) ~ 2^(24*n + 1/2) / (Gamma(1/4) * 7^(7*n + 1/4) * n^(3/4)).
%F A383965 From _Seiichi Manyama_, Aug 16 2025: (Start)
%F A383965 Sum_{k=0..n} a(k) * a(n-k) = A004381(n).
%F A383965 G.f.: 1/sqrt(1 - 8*x*g^7) where g = 1+x*g^8 is the g.f. of A007556.
%F A383965 G.f.: sqrt( g/(8-7*g) ) where g = 1+x*g^8 is the g.f. of A007556. (End)
%p A383965 a:= proc(n) option remember; `if`(n=0, 1,
%p A383965       (binomial(8*n, n)-add(a(j)*a(n-j), j=1..n-1))/2)
%p A383965     end:
%p A383965 seq(a(n), n=0..20);  # _Alois P. Heinz_, Jun 06 2025
%t A383965 nmax = 20; self = ConstantArray[0, nmax + 1]; self[[1]] = 1; self[[2]] = 4; Do[self[[k+1]] = (Binomial[8*k, k] - Sum[self[[j+1]]*self[[k - j + 1]], {j, 1, k-1}]) / (2*self[[1]]);, {k, 2, nmax}]; self
%Y A383965 Cf. A004381, A007556, A208977, A384695.
%K A383965 nonn
%O A383965 0,2
%A A383965 _Vaclav Kotesovec_, Jun 06 2025