|
%I #11 Mar 30 2012 18:37:28
%S 1,1,1,2,2,2,6,4,4,6,20,10,12,10,20,72,30,28,28,30,72,260,102,84,104,
%T 84,102,260,996,362,260,268,268,260,362,996,3772,1358,892,832,1144,
%U 832,892,1358,3772,14852,5130,3236,2928,2956,2956,2928,3236,5130,14852,58204,19982,12044,10072,9948,13736,9948,10072,12044,19982,58204
%N Symmetric triangle T, read by rows, where the matrix product of T and T transpose yields a square array which, when read by antidiagonals, equals this triangle read by rows.
%F T(n,k) = Sum_{j=0..k} T(n-k,j)*T(k,j) for n>0, k>=0, with T(0,0)=1.
%F Column 0 (A194950) equals row sums of triangle.
%F Central terms (A194951) equals sums of squares of terms in rows.
%e Triangle T begins:
%e 1;
%e 1, 1;
%e 2, 2, 2;
%e 6, 4, 4, 6;
%e 20, 10, 12, 10, 20;
%e 72, 30, 28, 28, 30, 72;
%e 260, 102, 84, 104, 84, 102, 260;
%e 996, 362, 260, 268, 268, 260, 362, 996;
%e 3772, 1358, 892, 832, 1144, 832, 892, 1358, 3772;
%e 14852, 5130, 3236, 2928, 2956, 2956, 2928, 3236, 5130, 14852;
%e 58204, 19982, 12044, 10072, 9948, 13736, 9948, 10072, 12044, 19982, 58204; ...
%e ...
%e Matrix product of T and T transpose, T*T~, yields the square array:
%e 1, 1, 2, 6, 20, 72, 260, 996, 3772, ...;
%e 1, 2, 4, 10, 30, 102, 362, 1358, 5130, ...;
%e 2, 4, 12, 28, 84, 260, 892, 3236, 12044, ...;
%e 6, 10, 28, 104, 268, 832, 2928, 10072, 36624, ...;
%e 20, 30, 84, 268, 1144, 2956, 9948, 34700, 130924, ...;
%e 72, 102, 260, 832, 2956, 13736, 36908, 124116, 454820, ...;
%e 260, 362, 892, 2928, 9948, 36908, 180936, 488748, 1693572, ...;
%e 996, 1358, 3236, 10072, 34700, 124116, 488748, 2524968, 6901788, ...;
%e 3772, 5130, 12044, 36624, 130924, 454820, 1693572, 6901788, 36428808, ...;
%e ...
%e which, when read by antidiagonals, equals this triangle read by rows.
%o (PARI) {T(n,k)=local(M=matrix(n+1,n+1,r,c,if(r>=c,1))); for(i=1,n,M=matrix(n+1,n+1,r,c,if(r>=c,if(c==1,if(r==1,1,sum(j=1,r-1,(M*M~)[r-j,j])), (M*M~)[r-c+1,c]))));M[n+1,k+1]}
%Y Cf. A194950, A194951.
%K nonn,tabl
%O 0,4
%A _Paul D. Hanna_, Sep 05 2011
|
