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%I #13 Apr 16 2016 04:32:58
%S 1,1,1,3,5,14,30,82,203,552,1458,4004,10956,30514,85259,240507,681571,
%T 1943472,5565744,16011492,46233297,133975566,389455910,1135431759,
%U 3319060758,9726061473,28565447104,84073146827,247924840773,732439856638,2167507140543,6424491527538,19070573498367,56688719414910,168733726744153,502859937709589,1500383417733522,4481672952197057,13400947416395067,40111136395590224
%N G.f. A(x) satisfies: A(x) = x + A( x^2 + x*A(x)^2 ).
%C Compare g.f. to: C(x) = x + C( x^2 + 2*x*C(x)^2 ) where C(x) = x + C(x)^2 is a g.f. of the Catalan numbers (A000108).
%H Paul D. Hanna, <a href="/A271867/b271867.txt">Table of n, a(n) for n = 1..300</a>
%F a(n) ~ c * d^n / n^(3/2), where d = 3.109781515236165... and c = 0.1963355843719... . - _Vaclav Kotesovec_, Apr 16 2016
%e G.f.: A(x) = x + x^2 + x^3 + 3*x^4 + 5*x^5 + 14*x^6 + 30*x^7 + 82*x^8 + 203*x^9 + 552*x^10 + 1458*x^11 + 4004*x^12 + 10956*x^13 + 30514*x^14 + 85259*x^15 +...
%e where A(x) = x + A( x^2 + x*A(x)^2 ).
%e RELATED SERIES.
%e A(x)^2 = x^2 + 2*x^3 + 3*x^4 + 8*x^5 + 17*x^6 + 44*x^7 + 107*x^8 + 282*x^9 + 739*x^10 + 1994*x^11 + 5414*x^12 + 14906*x^13 + 41374*x^14 + 115820*x^15 +...
%e Let G(x,y) = x + G(x^2 + x*y*G(x,y)^2, y), then the coefficient of x^n in G(x,y) begins:
%e n=1: 1;
%e n=2: 1;
%e n=3: y;
%e n=4: 1 + 2*y;
%e n=5: 3*y + 2*y^2;
%e n=6: 7*y + 7*y^2;
%e n=7: 4*y + 21*y^2 + 5*y^3;
%e n=8: 1 + 6*y + 46*y^2 + 29*y^3;
%e n=9: 9*y + 65*y^2 + 114*y^3 + 15*y^4;
%e n=10: 13*y + 113*y^2 + 304*y^3 + 122*y^4;
%e n=11: 8*y + 169*y^2 + 649*y^3 + 582*y^4 + 50*y^5;
%e n=12: 19*y + 229*y^2 + 1311*y^3 + 1931*y^4 + 514*y^5;
%e n=13: 14*y + 326*y^2 + 2289*y^3 + 5235*y^4 + 2915*y^5 + 177*y^6;
%e n=14: 4*y + 511*y^2 + 3800*y^3 + 12353*y^4 + 11667*y^5 + 2179*y^6;
%e n=15: 8*y + 528*y^2 + 6365*y^3 + 25663*y^4 + 37605*y^5 + 14439*y^6 + 651*y^7;
%e n=16: 1 + 14*y + 602*y^2 + 9933*y^3 + 50117*y^4 + 102960*y^5 + 67567*y^6 + 9313*y^7; ...
%e where the coefficients of x^n at y=2 yield the Catalan sequence (A000108)
%e and the coefficients of x^n at y=1 yield this sequence.
%o (PARI) {a(n) = my(A=x+x^2 +x*O(x^n)); for(i=1, n, A = x + subst(A, x, x^2 + x*A^2) ) ; polcoeff(A, n)}
%o for(n=1, 40, print1(a(n), ", "))
%Y Cf. A271868.
%K nonn
%O 1,4
%A _Paul D. Hanna_, Apr 15 2016
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