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A264231 G.f. satisfies: A(x) = A( -x/(1-4*x) ) / sqrt(1-4*x), with A(0)=1.

%I A264231 #15 Aug 10 2016 22:12:46
%S A264231 1,1,3,10,37,144,582,2418,10266,44353,194395,862308,3864100,17466055,
%T A264231 79537815,364543920,1680205044,7782321357,36202366293,169056409476,
%U A264231 792157776138,3723252477558,17548166926908,82913613567012,392650956009063,1863327440038659,8859299215070493,42196115735716894,201303626788647712,961803888607574670
%N A264231 G.f. satisfies: A(x)  =  A( -x/(1-4*x) ) / sqrt(1-4*x), with A(0)=1.
%C A264231 Self-convolution square root of A264224.
%C A264231 Radius of convergence is r = 1/5 with A(1/5) = sqrt(5).
%F A264231 G.f. A(x) satisfies:
%F A264231 (1) A(x)^2  =  A( x^2/(1-4*x) ) / sqrt(1-4*x).
%F A264231 (2) A( x/(1+2*x) )^2 / (1+2*x)  =  A( x^2/(1-4*x^2) ) / sqrt(1-4*x^2).
%e A264231 G.f.: A(x) = 1 + x + 3*x^2 + 10*x^3 + 37*x^4 + 144*x^5 + 582*x^6 + 2418*x^7 +...
%e A264231 where A(x) = A( -x/(1-4*x) ) / sqrt(1-4*x).
%e A264231 RELATED SERIES.
%e A264231 Note that A(x)^2 = A( x^2/(1-4*x) ) / sqrt(1-4*x):
%e A264231 A(x)^2 = 1 + 2*x + 7*x^2 + 26*x^3 + 103*x^4 + 422*x^5 + 1774*x^6 + 7604*x^7 + 33109*x^8 + 146042*x^9 + 651256*x^10 +...+ A264224(n-1)*x^n +...
%e A264231 where
%e A264231 A( x^2/(1-4*x) ) = 1 + x^2 + 4*x^3 + 19*x^4 + 88*x^5 + 410*x^6 + 1912*x^7 + 8933*x^8 + 41808*x^9 + 196016*x^10 + 920640*x^11 + 4331590*x^12 +...
%e A264231 Also,
%e A264231 A( x/(1+2*x) ) = 1 + x + x^2 + 2*x^3 + 5*x^4 + 8*x^5 + 30*x^6 + 42*x^7 + 198*x^8 + 257*x^9 + 1385*x^10 + 1724*x^11 + 10072*x^12 +...
%e A264231 such that
%e A264231 A( x/(1+2*x) )^2 / (1+2*x) = 1 + 3*x^2 + 15*x^4 + 90*x^6 + 597*x^8 + 4212*x^10 + 30942*x^12 + 233766*x^14 + 1802706*x^16 +...
%e A264231 equals A( x^2/(1-4*x^2) ) / sqrt(1-4*x^2).
%o A264231 (PARI) {a(n) = my(A=1+x,X=x+x*O(x^n)); for(i=1, n, A = subst(A, x, x^2/(1-4*X))^(1/2)/(1-4*X)^(1/4)  ); polcoeff(A, n)}
%o A264231 for(n=0,30, print1(a(n),", "))
%Y A264231 Cf. A264224.
%K A264231 nonn
%O A264231 0,3
%A A264231 _Paul D. Hanna_, Nov 29 2015