A086620 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-y) + xy*f(x,y)^2.

1, 1, 1, 1, 3, 1, 1, 5, 5, 1, 1, 7, 14, 7, 1, 1, 9, 28, 28, 9, 1, 1, 11, 47, 79, 47, 11, 1, 1, 13, 71, 175, 175, 71, 13, 1, 1, 15, 100, 331, 504, 331, 100, 15, 1, 1, 17, 134, 562, 1196, 1196, 562, 134, 17, 1, 1, 19, 173, 883, 2464, 3514, 2464, 883, 173, 19, 1, 1, 21, 217

Offset: 0

Author

Paul D. Hanna, Jul 24 2003

Keywords

• nonn
• tabl

Comments

Determinants of upper left n X n matrices results in A086619: {1,2,10,150,7650,1438200,1051324200,...}, which is the products of the first n terms of the binomial transform of Catalan numbers (A007317): {1,2,5,15,51,188,731,2950,...}.

Examples

			Rows begin:
1,_1,__1,__1,___1,____1,____1,_____1, ...
1,_3,__5,__7,___9,___11,___13,____15, ...
1,_5,_14,_28,__47,___71,__100,___134, ...
1,_7,_28,_79,_175,__331,__562,___883, ...
1,_9,_47,175,_504,_1196,_2464,__4572, ...
1,11,_71,331,1196,_3514,_8764,_19244, ...
1,13,100,562,2464,_8764,26172,_67740, ...
1,15,134,883,4572,19244,67740,204831, ...
		

Crossrefs

Cf. A086621 (diagonal), A086622 (antidiagonal sums), A086619 (determinants).

Formula

Contribution from Paul Barry, Feb 04 2009: (Start)

T(n,k)=sum{j=0..n+k, C(k,j-k)*C(n+2k-j,k)*if(k<=j,A000108(n-k),0)};

Regarded as a number triangle read by row, columns are generated by sum{j=0..k, C(k,j)*A000108(j)*x^j}*x^k/(1-x)^(k+1). (End)