%I #14 Nov 30 2014 22:07:58

%S 1,1,1,3,7,19,59,179,615,2141,7853,30179,118663,487581,2050325,

%T 8890743,39593811,180367655,842738375,4018170415,19578229355,

%U 97275623919,492406520567,2538806377419,13312298342703,70986151599775,384532625928335,2115147900599015

%N G.f. A(x) satisfies A(x) = 1 + x/sqrt(1-4*x^2) * A(x/sqrt(1-4*x^2)).

%F a(n) = sum(m=0..(n-1)/2, a(n-2*m-1) * binomial(n/2-1,m) * 2^(2*m)), n>0, a(0)=1.

%F G.f.: Sum_{n>=0} x^n / Product_{k=0..n} sqrt(1 - 4*k*x^2). - _Paul D. Hanna_, Nov 30 2014

%F G.f. A(x) satisfies: A(x/sqrt(1+4*x^2)) = 1 + x*A(x). - _Paul D. Hanna_, Nov 30 2014

%e G.f.: A(x) = = 1 + x + x^2 + 3*x^3 + 7*x^4 + 19*x^5 + 59*x^6 + 179*x^7 + ...

%e such that A(x) = 1 + G(x)*A(G(x)) where

%e G(x) = x/sqrt(1-4*x^2) = x + 2*x^3 + 6*x^5 + 20*x^7 + 70*x^9 + 252*x^11 + ... .

%e The g.f. also equals the series

%e A(x) = 1 + x/sqrt(1-4*x^2) + x^2/sqrt((1-4*x^2)*(1-8*x^2)) + x^3/sqrt((1-4*x^2)*(1-8*x^2)*(1-12*x^2)) + x^4/sqrt((1-4*x^2)*(1-8*x^2)*(1-12*x^2)*(1-16*x^2)) + ... .

%o (Maxima)

%o a(n):=if n=0 then 1 else sum(a(n-2*m-1)*binomial(n/2-1,m)*2^(2*m),m,0,(n-1)/2);

%o (PARI) {a(n)=polcoeff( sum(m=0,n, x^m / prod(k=0,m, sqrt(1 - 4*k*x^2 +x*O(x^n)) )), n)}

%o for(n=0,30,print1(a(n),", ")) \\ _Paul D. Hanna_, Nov 30 2014

%K nonn

%O 0,4

%A _Vladimir Kruchinin_, Nov 28 2011

%E Name corrected slightly by _Paul D. Hanna_, Nov 30 2014