This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A108553 #6 Jun 13 2017 02:02:55
%S A108553 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 A108553 1,61,411,625,309,61,13,1,1,85,853,2051,1681,561,85,15,1,1,113,1583,
%U A108553 5577,6981,3721,923,113,17,1,1,145,2705,13203,23673,18733,7225,1415,145,19,1
%N A108553 Square array, read by antidiagonals, where row n equals the crystal ball sequence for D_n lattice.
%C A108553 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 A108553 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 A108553 Square array begins:
%e A108553 1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,...
%e A108553 1,3,5,7,9,11,13,15,17,19,21,23,25,27,...
%e A108553 1,5,13,25,41,61,85,113,145,181,221,265,...
%e A108553 1,13,55,147,309,561,923,1415,2057,2869,...
%e A108553 1,25,169,625,1681,3721,7225,12769,21025,...
%e A108553 1,41,411,2051,6981,18733,42783,86983,...
%e A108553 1,61,853,5577,23673,76389,204205,476113,...
%e A108553 1,85,1583,13203,68853,264825,824083,...
%e A108553 Inverse binomial transform of rows gives
%e A108553 rows of triangle A108556:
%e A108553 1,
%e A108553 1,2,
%e A108553 1,4,4,
%e A108553 1,12,30,20,
%e A108553 1,24,120,192,96,
%e A108553 1,40,330,940,1080,432, ...
%e A108553 Product of the g.f. of row n and (1-x)^(n+1)
%e A108553 generates the symmetric triangle A108558:
%e A108553 1;
%e A108553 1,1;
%e A108553 1,2,1;
%e A108553 1,9,9,1;
%e A108553 1,20,54,20,1;
%e A108553 1,35,180,180,35,1; ...
%e A108553 The row sums of triangle A108558 equals the
%e A108553 main diagonal of triangle A108556.
%o A108553 (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 A108553 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 A108553 nonn,tabl
%O A108553 0,5
%A A108553 _Paul D. Hanna_, Jun 10 2005