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%I #6 Jun 13 2017 02:02:55
%S 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,
%T 1,61,411,625,309,61,13,1,1,85,853,2051,1681,561,85,15,1,1,113,1583,
%U 5577,6981,3721,923,113,17,1,1,145,2705,13203,23673,18733,7225,1415,145,19,1
%N Square array, read by antidiagonals, where row n equals the crystal ball sequence for D_n lattice.
%C 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.
%F 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.
%e Square array begins:
%e 1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,...
%e 1,3,5,7,9,11,13,15,17,19,21,23,25,27,...
%e 1,5,13,25,41,61,85,113,145,181,221,265,...
%e 1,13,55,147,309,561,923,1415,2057,2869,...
%e 1,25,169,625,1681,3721,7225,12769,21025,...
%e 1,41,411,2051,6981,18733,42783,86983,...
%e 1,61,853,5577,23673,76389,204205,476113,...
%e 1,85,1583,13203,68853,264825,824083,...
%e Inverse binomial transform of rows gives
%e rows of triangle A108556:
%e 1,
%e 1,2,
%e 1,4,4,
%e 1,12,30,20,
%e 1,24,120,192,96,
%e 1,40,330,940,1080,432, ...
%e Product of the g.f. of row n and (1-x)^(n+1)
%e generates the symmetric triangle A108558:
%e 1;
%e 1,1;
%e 1,2,1;
%e 1,9,9,1;
%e 1,20,54,20,1;
%e 1,35,180,180,35,1; ...
%e The row sums of triangle A108558 equals the
%e main diagonal of triangle A108556.
%o (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))))))
%Y 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).
%K nonn,tabl
%O 0,5
%A _Paul D. Hanna_, Jun 10 2005
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