A108553 Square array, read by antidiagonals, where row n equals the crystal ball sequence for D_n lattice.
1, 1, 1, 1, 3, 1, 1, 5, 5, 1, 1, 13, 13, 7, 1, 1, 25, 55, 25, 9, 1, 1, 41, 169, 147, 41, 11, 1, 1, 61, 411, 625, 309, 61, 13, 1, 1, 85, 853, 2051, 1681, 561, 85, 15, 1, 1, 113, 1583, 5577, 6981, 3721, 923, 113, 17, 1, 1, 145, 2705, 13203, 23673, 18733, 7225, 1415, 145, 19, 1
Offset: 0
Examples
Square array begins: 1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,... 1,3,5,7,9,11,13,15,17,19,21,23,25,27,... 1,5,13,25,41,61,85,113,145,181,221,265,... 1,13,55,147,309,561,923,1415,2057,2869,... 1,25,169,625,1681,3721,7225,12769,21025,... 1,41,411,2051,6981,18733,42783,86983,... 1,61,853,5577,23673,76389,204205,476113,... 1,85,1583,13203,68853,264825,824083,... Inverse binomial transform of rows gives rows of triangle A108556: 1, 1,2, 1,4,4, 1,12,30,20, 1,24,120,192,96, 1,40,330,940,1080,432, ... Product of the g.f. of row n and (1-x)^(n+1) generates the symmetric triangle A108558: 1; 1,1; 1,2,1; 1,9,9,1; 1,20,54,20,1; 1,35,180,180,35,1; ... The row sums of triangle A108558 equals the main diagonal of triangle A108556.
Crossrefs
Programs
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PARI
T(n,k)=if(n<0 || k<0,0,if(n==0 || k==0,1,if(n==1,2*k+1, sum(j=0,k,binomial(n+k-j,k-j)*(binomial(2*n,2*j)-2*n*binomial(n-2,j-1))))))
Formula
T(n, k) = Sum_{j=0..n} C(n+k-j, k-j)*[C(2*n, 2*j) - 2*j*(n-j)*C(n, j)/(n-1)] for n>1, with T(0, k)=1, T(1, k)=2*k+1.
Comments