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A222050 G.f. satisfies: A(x) = sqrt(1 + 2*x*A(x)^4 + 3*x^2*A(x)^6).

%I A222050 #8 Feb 06 2013 20:03:31
%S A222050 1,1,5,30,209,1573,12478,102714,869193,7514445,66083025,589294500,
%T A222050 5316256278,48431659786,444928748618,4117185679310,38340948482745,
%U A222050 359047299072777,3379057486089649,31942315551724102,303158909307122141,2887629443604011421,27595392738011189028
%N A222050 G.f. satisfies: A(x) = sqrt(1 + 2*x*A(x)^4 + 3*x^2*A(x)^6).
%F A222050 G.f.: sqrt( (1/x)*Series_Reversion( x*(1-2*x-3*x^2) ) ).
%F A222050 a(n) = [x^n] sqrt( 1/(1-2*x-3*x^2)^(2*n+1) ) / (2*n+1).
%F A222050 a(n) = A222052(n)/(2*n+1).
%e A222050 G.f.: A(x) = 1 + x + 5*x^2 + 30*x^3 + 209*x^4 + 1573*x^5 + 12478*x^6 +...
%e A222050 Related expansions.
%e A222050 A(x)^2 = 1 + 2*x + 11*x^2 + 70*x^3 + 503*x^4 + 3864*x^5 + 31092*x^6 +...
%e A222050 A(x)^4 = 1 + 4*x + 26*x^2 + 184*x^3 + 1407*x^4 + 11280*x^5 + 93606*x^6 +...
%e A222050 A(x)^6 = 1 + 6*x + 45*x^2 + 350*x^3 + 2844*x^4 + 23814*x^5 + 204149*x^6 +...
%e A222050 where A(x)^2 = 1 + 2*x*A(x)^4 + 3*x^2*A(x)^6.
%e A222050 Let G(x) = 1/sqrt(1-2*x-3*x^2) denote the g.f. of A002426,
%e A222050 then the array of coefficients of x^k in G(x)^(2*n+1) begins:
%e A222050 G(x)^1 : [1,  1,   3,    7,    19,    51,    141,     393,...];
%e A222050 G(x)^3 : [1,  3,  12,   40,   135,   441,   1428,    4572,...];
%e A222050 G(x)^5 : [1,  5,  25,  105,   420,  1596,   5880,   21120,...];
%e A222050 G(x)^7 : [1,  7,  42,  210,   966,  4158,  17094,   67782,...];
%e A222050 G(x)^9 : [1,  9,  63,  363,  1881,  9009,  40755,  176319,...];
%e A222050 G(x)^11: [1, 11,  88,  572,  3289, 17303,  85228,  398684,...];
%e A222050 G(x)^13: [1, 13, 117,  845,  5330, 30498, 162214,  814606,...];
%e A222050 G(x)^15: [1, 15, 150, 1190,  8160, 50388, 287470, 1540710,...]; ...
%e A222050 in which the main diagonal (A222052) forms this sequence like so:
%e A222050 [1/1, 3/3, 25/5, 210/7, 1881/9, 17303/11, 162214/13, 1540710/15,...].
%o A222050 (PARI) {a(n)=polcoeff(sqrt(1/x*serreverse(x*(1-2*x-3*x^2)+x^2*O(x^n))),n)}
%o A222050 for(n=0,25,print1(a(n),", "))
%o A222050 (PARI) {a(n)=polcoeff(1/sqrt(1-2*x-3*x^2+x*O(x^n))^(2*n+1),n)/(2*n+1)}
%o A222050 for(n=0,25,print1(a(n),", "))
%Y A222050 Cf. A222051, A222052, A002426.
%K A222050 nonn
%O A222050 0,3
%A A222050 _Paul D. Hanna_, Feb 06 2013