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

Author

Paul D. Hanna, Jun 10 2005

Keywords

• nonn
• tabl

Comments

Rows 0 and 2 are included by extension since they fit the formula. Row 1 equals the odd numbers in order that triangle A108556 maintains that A108556(n,n-1) = (n/2)*A108556(n,n) for all n>=1, where row n of triangle A108556 equals the inverse binomial transform of row n of this square array.

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

Cf. A108554 (diagonal), A108555 (antidiagonal sums), A108556, A108558, A001844 (row 2), A005902 (row 3), A007204 (row 4), A008356 (row 5), A008358 (row 6), A008360 (row 7), A008362 (row 8), A008377 (row 9), A008379 (row 10).

Programs

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.