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A208977 Self-convolution square-root of A005810, where A005810(n) = binomial(4*n,n).

%I A208977 #16 Jun 07 2025 08:12:32
%S A208977 1,2,12,86,666,5388,44832,380424,3275172,28512248,250413856,
%T A208977 2215112886,19711078686,176276723508,1583186541144,14271487891512,
%U A208977 129063176166570,1170480053359908,10641805703955624,96970507481607972,885397365149468076,8098908925136867112
%N A208977 Self-convolution square-root of A005810, where A005810(n) = binomial(4*n,n).
%F A208977 G.f.: A(x) = sqrt( G(x)/(4 - 3*G(x)) ) where G(x) = 1 + x*G(x)^4 is the g.f. of A002293. [From a formula by Mark van Hoeij in A005810]
%F A208977 From _Vaclav Kotesovec_, Jun 06 2025: (Start)
%F A208977 Recurrence: 81*(n-1)*n*(2*n - 3)*(3*n - 2)*(3*n - 1)*a(n) = 24*(n-1)*(1152*n^4 - 4608*n^3 + 6698*n^2 - 4180*n + 915)*a(n-1) - 16*(2*n - 1)*(8*n - 15)*(8*n - 13)*(8*n - 11)*(8*n - 9)*a(n-2).
%F A208977 a(n) ~ 2^(8*n + 1/4) / (Gamma(1/4) * n^(3/4) * 3^(3*n + 1/4)) * (1 - Gamma(1/4)^2 / (24*Pi*sqrt(3*n))). (End)
%e A208977 G.f.: A(x) = 1 + 2*x + 12*x^2 + 86*x^3 + 666*x^4 + 5388*x^5 +...
%e A208977 The square of the g.f. equals the g.f. of A005810:
%e A208977 A(x)^2 = 1 + 4*x + 28*x^2 + 220*x^3 + 1820*x^4 + 15504*x^5 +...
%e A208977 The g.f. of A002293 is G(x) = 1 + x*G(x)^4:
%e A208977 G(x) = 1 + x + 4*x^2 + 22*x^3 + 140*x^4 + 969*x^5 + 7084*x^6 +...
%p A208977 a:= proc(n) option remember; `if`(n=0, 1,
%p A208977       (binomial(4*n, n)-add(a(j)*a(n-j), j=1..n-1))/2)
%p A208977     end:
%p A208977 seq(a(n), n=0..21);  # _Alois P. Heinz_, Jun 06 2025
%t A208977 nmax = 20; self = ConstantArray[0, nmax + 1]; self[[1]] = 1; self[[2]] = 2; Do[self[[k+1]] = (Binomial[4*k, k] - Sum[self[[j+1]]*self[[k-j+1]], {j, 1, k-1}]) / (2*self[[1]]);, {k, 2, nmax}]; self (* _Vaclav Kotesovec_, Jun 06 2025 *)
%o A208977 (PARI) {a(n)=polcoeff(sum(k=0,n,binomial(4*k,k)*x^k +x*O(x^n))^(1/2),n)}
%o A208977 for(n=0,41,print1(a(n),", "))
%Y A208977 Cf. A005810, A002293, A383965.
%K A208977 nonn
%O A208977 0,2
%A A208977 _Paul D. Hanna_, Mar 03 2012