%I #8 Feb 06 2013 20:03:31
%S 1,1,5,30,209,1573,12478,102714,869193,7514445,66083025,589294500,
%T 5316256278,48431659786,444928748618,4117185679310,38340948482745,
%U 359047299072777,3379057486089649,31942315551724102,303158909307122141,2887629443604011421,27595392738011189028
%N G.f. satisfies: A(x) = sqrt(1 + 2*x*A(x)^4 + 3*x^2*A(x)^6).
%F G.f.: sqrt( (1/x)*Series_Reversion( x*(1-2*x-3*x^2) ) ).
%F a(n) = [x^n] sqrt( 1/(1-2*x-3*x^2)^(2*n+1) ) / (2*n+1).
%F a(n) = A222052(n)/(2*n+1).
%e G.f.: A(x) = 1 + x + 5*x^2 + 30*x^3 + 209*x^4 + 1573*x^5 + 12478*x^6 +...
%e Related expansions.
%e A(x)^2 = 1 + 2*x + 11*x^2 + 70*x^3 + 503*x^4 + 3864*x^5 + 31092*x^6 +...
%e A(x)^4 = 1 + 4*x + 26*x^2 + 184*x^3 + 1407*x^4 + 11280*x^5 + 93606*x^6 +...
%e A(x)^6 = 1 + 6*x + 45*x^2 + 350*x^3 + 2844*x^4 + 23814*x^5 + 204149*x^6 +...
%e where A(x)^2 = 1 + 2*x*A(x)^4 + 3*x^2*A(x)^6.
%e Let G(x) = 1/sqrt(1-2*x-3*x^2) denote the g.f. of A002426,
%e then the array of coefficients of x^k in G(x)^(2*n+1) begins:
%e G(x)^1 : [1, 1, 3, 7, 19, 51, 141, 393,...];
%e G(x)^3 : [1, 3, 12, 40, 135, 441, 1428, 4572,...];
%e G(x)^5 : [1, 5, 25, 105, 420, 1596, 5880, 21120,...];
%e G(x)^7 : [1, 7, 42, 210, 966, 4158, 17094, 67782,...];
%e G(x)^9 : [1, 9, 63, 363, 1881, 9009, 40755, 176319,...];
%e G(x)^11: [1, 11, 88, 572, 3289, 17303, 85228, 398684,...];
%e G(x)^13: [1, 13, 117, 845, 5330, 30498, 162214, 814606,...];
%e G(x)^15: [1, 15, 150, 1190, 8160, 50388, 287470, 1540710,...]; ...
%e in which the main diagonal (A222052) forms this sequence like so:
%e [1/1, 3/3, 25/5, 210/7, 1881/9, 17303/11, 162214/13, 1540710/15,...].
%o (PARI) {a(n)=polcoeff(sqrt(1/x*serreverse(x*(1-2*x-3*x^2)+x^2*O(x^n))),n)}
%o for(n=0,25,print1(a(n),", "))
%o (PARI) {a(n)=polcoeff(1/sqrt(1-2*x-3*x^2+x*O(x^n))^(2*n+1),n)/(2*n+1)}
%o for(n=0,25,print1(a(n),", "))
%Y Cf. A222051, A222052, A002426.
%K nonn
%O 0,3
%A _Paul D. Hanna_, Feb 06 2013