A102916 Triangle, read by rows, where the antidiagonals are formed by interleaving the rows of triangle A102098 with the rows of its matrix square (A102920).
1, 1, 2, 1, 4, 3, 3, 8, 9, 4, 7, 40, 27, 16, 5, 36, 152, 189, 64, 25, 6, 139, 1128, 999, 576, 125, 36, 7, 1036, 6200, 9720, 3904, 1375, 216, 49, 8, 5711, 61120, 69687, 47040, 11375, 2808, 343, 64, 9, 56355, 442552, 857466, 416704, 163500, 27432, 5145, 512
Offset: 0
Examples
Rows begin: [1], [1,2], [1,4,3], [3,8,9,4], [7,40,27,16,5], [36,152,189,64,25,6], [139,1128,999,576,125,36,7], [1036,6200,9720,3904,1375,216,49,8], [5711,61120,69687,47040,11375,2808,343,64,9],... The antidiagonals are formed by interleaving the rows of triangle A102098: [1], [1,2], [7,8,3], [139,152,27,4],... with the rows of the matrix square of A102098, which is triangle A102920: [1], [3,4], [36,40,9], [1036,1128,189,16],... G.f. for Column 0 (A102917): 1 = 1*(1-x) + 1*x*(1-x) + 1*x^2*(1-x)(1-2x) + 3*x^3*(1-x)(1-2x) + 7*x^4*(1-x)(1-2x)(1-3x) + 36*x^5*(1-x)(1-2x)(1-3x) +... + A082162(n)*x^(2n)*(1-x)(1-2x)*..*(1-(n+1)x) + A102921(n)*x^(2n+1)*(1-x)(1-2x)*..*(1-(n+1)x) + ... G.f. for Column 1 (A102918): 2 = 2*(1-2x) + 4*x*(1-2x) + 8*x^2*(1-2x)(1-3x) + 40*x^3*(1-2x)(1-3x) + 152*x^4*(1-2x)(1-3x)(1-4x) + 1128*x^5*(1-2x)(1-3x)(1-4x) +... + T(2n+1,1)*x^(2n)*(1-2x)(1-3x)*..*(1-(n+2)x) + T(2n+2,1)*x^(2n+1)*(1-2x)(1-3x)*..*(1-(n+2)x) + ...
Programs
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PARI
{T(n,k)=if(n
Formula
G.f. for column k: T(k, k) = k+1 = Sum_{n>=0} T(n+k, k)*x^n*Product_{j=k..[n/2+k]} (1-(j+1)*x).
Comments