A086626 - OEIS

A086626

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-xy)/[(1-x)(1-y)] + xy*f(x,y)^3.

%I #7 Nov 15 2019 17:42:14

%S 1,1,1,1,1,1,1,3,3,1,1,6,9,6,1,1,10,24,24,10,1,1,15,57,84,57,15,1,1,

%T 21,120,249,249,120,21,1,1,28,228,654,907,654,228,28,1,1,36,399,1554,

%U 2880,2880,1554,399,36,1,1,45,654,3384,8178,10821,8178,3384,654,45,1,1,55,1017,6831,21156,35949,35949,21156,6831,1017,55,1

%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-xy)/[(1-x)(1-y)] + xy*f(x,y)^3.

%e Rows begin:

%e 1,_1,__1,___1,____1,_____1,_____1,______1, ...

%e 1,_1,__3,___6,___10,____15,____21,_____28, ...

%e 1,_3,__9,__24,___57,___120,___228,____399, ...

%e 1,_6,_24,__84,__249,___654,__1554,___3384, ...

%e 1,10,_57,_249,__907,__2880,__8178,__21156, ...

%e 1,15,120,_654,_2880,_10821,_35949,_107952, ...

%e 1,21,228,1554,_8178,_35949,137832,_473331, ...

%e 1,28,399,3384,21156,107952,473331,1840560, ...

%t m = 12; f[_, _] = 0;

%t Do[f[x_, y_] = (1 - x y)/((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. A086627 (diagonal), A086628 (antidiagonal sums).

%K nonn,tabl

%O 0,8

%A _Paul D. Hanna_, Jul 24 2003