通用公式:A(x,y)=1+x*y+x^2*(-2*y+2*y^2)+x^3*(3*y-6*y^2+5*y^3)+x ^4*(-6*y+14*y^2-20*y^3+14*y^4)+x*5*(11*y-36*y^2+59*y^3-70*y^4+42*y^5)+x|6*(-18*y+87*y^2-176*y^3+246*y^4-252*y^5+132*y^6)+x^7*(28*y-190*y^2+500*y^3-824*y^4+1022*y^5-924*y ^6+429*y^7)+x^8*(-44*y+386*y^2-1312*y^3+2615*y^4-3780*y^5+4236*y^6-3432*y^7+1430*y^8)+x^9*(69*y-756*y^2+3218*y^3-7734*y^4+13107*y^5-17112*y^6+17523*y^7-12870*y^8+4862*y^9)+。。。
其中A=A(x,y)满足
0=Sum_{n=-oo..+oo}x^n*A^n*(y-x^(n-1))^(n+1);
明确地,
0=((-A+1)/A)/x+y+(A*y^2-2*A*y+((A^3-1)/A^2))*x+A^2*y^3*x^2+(A^3*y^4-3*A^2*y^2)*x^3+(A^4*y^5+((3*A^4-1)/A^2)*y)*x^4+(A^5*y^6-4*A^3*y^3+((-A^5+1)/A^3))*x^5+A^6*y^7*x^6+(A^7*y^8-5*A^4*y^4+((6*A^5-1)/A^2)*y^2)*x^7+A^8*y^9*x^8+(A^9*y^10-6*A^5*y^5+(-4*A^6+2)/A^3)*y)*x^9+(A^10*y^11+((10*A^6-1)/A^2)*y^3)*x^10+。。。
A(x,y)中x^n*y^k系数的三角形开始于:
1;
0, 1;
0, -2, 2;
0, 3, -6, 5;
0, -6, 14, -20, 14;
0, 11, -36, 59, -70, 42;
0, -18, 87, -176, 246, -252, 132;
0, 28, -190, 500, -824, 1022, -924, 429;
0, -44, 386, -1312, 2615, -3780, 4236, -3432, 1430;
0, 69, -756, 3218, -7734, 13107, -17112, 17523, -12870, 4862;
0, -104, 1443, -7514, 21496, -42444, 64031, -76692, 72358, -48620, 16796;
0, 152, -2668, 16862, -56856, 129425, -223458, 307189, -340912, 298298, -184756, 58786;
0, -222, 4782, -36456, 144159, -375618, 734310, -1143924, 1453221, -1504932, 1227876, -705432, 208012; ...
其中主对角线等于加泰罗尼亚数字(A000108美元),列1等于Product_{n>=1}(1-q^(2*n-1))^2/(1-q ^(2*n))^2中的系数(A274621型).