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Symmetric square table of coefficients, read by antidiagonals, where T(n,k) is the coefficient of x^n*y^k in f(x,y) that satisfies f(x,y) = 1/[(1-x)(1-y)] + xy*f(x,y)^3.
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%I #7 Nov 15 2019 17:41:54

%S 1,1,1,1,2,1,1,4,4,1,1,7,13,7,1,1,11,34,34,11,1,1,16,76,124,76,16,1,1,

%T 22,151,370,370,151,22,1,1,29,274,952,1419,952,274,29,1,1,37,463,2185,

%U 4573,4573,2185,463,37,1,1,46,739,4579,12892,18037,12892,4579,739,46,1

%N Symmetric square table of coefficients, read by antidiagonals, where T(n,k) is the coefficient of x^n*y^k in f(x,y) that satisfies f(x,y) = 1/[(1-x)(1-y)] + xy*f(x,y)^3.

%C If 1 is subtracted from every element of the table, the resulting table forms the coefficients of f(x,y)^3, where f(x,y) = 1/[(1-x)(1-y)] + xy*f(x,y)^3.

%t m = 11; f[_, _] = 0;

%t Do[f[x_, y_] = 1/((1 - x)(1 - y)) + x y f[x, y]^3 + O[x]^m, {m}];

%t T =CoefficientList[# + O[y]^m, y]& /@ CoefficientList[f[x, y], x];

%t Table[T[[n-k+1, k]], {n, 1, m}, {k, 1, n}] // Flatten (* _Jean-François Alcover_, Nov 15 2019 *)

%Y Cf. A086630 (diagonal), A086631 (antidiagonal sums).

%K nonn,tabl

%O 0,5

%A _Paul D. Hanna_, Jul 27 2003