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G.f. satisfies: A(x,y) = 1-x + y*x + Series_Reversion( x/A(x,y)^2 ).
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%I #5 Feb 24 2019 01:13:57

%S 1,0,1,0,2,0,0,4,5,0,0,8,38,14,0,0,16,184,262,42,0,0,32,720,2460,1602,

%T 132,0,0,64,2480,16360,25837,9260,429,0,0,128,7840,87920,268134,

%U 237870,52040,1430,0,0,256,23296,408128,2109040,3638386,2023992,288494,4862,0,0,512,66048,1701504,13676128,40049492,43815744,16394336,1590638,16796,0

%N G.f. satisfies: A(x,y) = 1-x + y*x + Series_Reversion( x/A(x,y)^2 ).

%C The rightmost nonzero numbers in this triangle form the Catalan numbers (A000108).

%F G.f. A(x,y) satisfies: A(x/A(x,y)^2, y) = 1+x + (y-1)*x/A(x,y)^2.

%e This triangle begins:

%e 1;

%e 0, 1;

%e 0, 2, 0;

%e 0, 4, 5, 0;

%e 0, 8, 38, 14, 0;

%e 0, 16, 184, 262, 42, 0;

%e 0, 32, 720, 2460, 1602, 132, 0;

%e 0, 64, 2480, 16360, 25837, 9260, 429, 0;

%e 0, 128, 7840, 87920, 268134, 237870, 52040, 1430, 0;

%e 0, 256, 23296, 408128, 2109040, 3638386, 2023992, 288494, 4862, 0;

%e 0, 512, 66048, 1701504, 13676128, 40049492, 43815744, 16394336, 1590638, 16796, 0;

%e 0, 1024, 180480, 6531840, 76845728, 349863976, 653001202, 487491424, 128720399, 8765044, 58786, 0;

%e 0, 2048, 478720, 23485440, 386423488, 2571281744, 7476451420, 9591548748, 5139351752, 991185638, 48412190, 208012, 0; ...

%e Row sums (A120970) begin:

%e [1, 1, 2, 9, 60, 504, 4946, 54430, 655362, 8496454, 117311198, ...],

%e the g.f. of which satisfies: G(x) = 1 + Series_Reversion(x/G(x)^2).

%e GENERATING FUNCTION.

%e G.f.: A(x,y) = 1 + x*y + x^2*(2*y) + x^3*(4*y + 5*y^2) +

%e x^4*(8*y + 38*y^2 + 14*y^3) +

%e x^5*(16*y + 184*y^2 + 262*y^3 + 42*y^4) +

%e x^6*(32*y + 720*y^2 + 2460*y^3 + 1602*y^4 + 132*y^5) +

%e x^7*(64*y + 2480*y^2 + 16360*y^3 + 25837*y^4 + 9260*y^5 + 429*y^6) +

%e x^8*(128*y + 7840*y^2 + 87920*y^3 + 268134*y^4 + 237870*y^5 + 52040*y^6 + 1430*y^7) +

%e x^9*(256*y + 23296*y^2 + 408128*y^3 + 2109040*y^4 + 3638386*y^5 + 2023992*y^6 + 288494*y^7 + 4862*y^8) +...

%e where

%e A(x,y) = 1-x + y*x + Series_Reversion( x/A(x,y)^2 ).

%e RELATED SERIES.

%e A(x/A(x,y)^2, y) = 1 + y*x + (-2*y^2 + 2*y)*x^2 +

%e (3*y^3 - 7*y^2 + 4*y)*x^3 +

%e (-4*y^4 + 6*y^3 - 10*y^2 + 8*y)*x^4 +

%e (5*y^5 - 27*y^4 - 18*y^3 + 24*y^2 + 16*y)*x^5 +

%e (-6*y^6 - 14*y^5 - 312*y^4 + 60*y^3 + 240*y^2 + 32*y)*x^6 +

%e (7*y^7 - 147*y^6 - 1745*y^5 - 1675*y^4 + 2360*y^3 + 1136*y^2 + 64*y)*x^7 +

%e (-8*y^8 - 348*y^7 - 10744*y^6 - 25146*y^5 + 10246*y^4 + 21616*y^3 + 4256*y^2 + 128*y)*x^8 +

%e (9*y^9 - 1361*y^8 - 60738*y^7 - 267656*y^6 - 84094*y^5 + 265552*y^4 + 133952*y^3 + 14080*y^2 + 256*y)*x^9 +...

%o (PARI) {T(n,k) = local(A=[1]);for(i=1,n, A=Vec(1 + (y-1)*x + serreverse(x/Ser(A)^2))); polcoeff(A[n+1],k,y)}

%o for(n=0,10,for(k=0,n, print1(T(n,k),", "));print(""))

%Y Cf. A120970, A000108.

%K nonn

%O 0,5

%A _Paul D. Hanna_, May 10 2015