A194949 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.

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, 84, 102, 260, 996, 362, 260, 268, 268, 260, 362, 996, 3772, 1358, 892, 832, 1144, 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

Offset

0,4

Links

Table of n, a(n) for n=0..65.

Formula

T(n,k) = Sum_{j=0..k} T(n-k,j)*T(k,j) for n>0, k>=0, with T(0,0)=1.

Column 0 (A194950) equals row sums of triangle.

Central terms (A194951) equals sums of squares of terms in rows.

Example

Triangle T begins:
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, 84, 102, 260;
996, 362, 260, 268, 268, 260, 362, 996;
3772, 1358, 892, 832, 1144, 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; ...
...
Matrix product of T and T transpose, T*T~, yields the square array:
1, 1, 2, 6, 20, 72, 260, 996, 3772, ...;
1, 2, 4, 10, 30, 102, 362, 1358, 5130, ...;
2, 4, 12, 28, 84, 260, 892, 3236, 12044, ...;
6, 10, 28, 104, 268, 832, 2928, 10072, 36624, ...;
20, 30, 84, 268, 1144, 2956, 9948, 34700, 130924, ...;
72, 102, 260, 832, 2956, 13736, 36908, 124116, 454820, ...;
260, 362, 892, 2928, 9948, 36908, 180936, 488748, 1693572, ...;
996, 1358, 3236, 10072, 34700, 124116, 488748, 2524968, 6901788, ...;
3772, 5130, 12044, 36624, 130924, 454820, 1693572, 6901788, 36428808, ...;
...
which, when read by antidiagonals, equals this triangle read by rows.

Prog

(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]}

Crossrefs

Cf. A194950, A194951.

Sequence in context: A209752 A119918 A084867 * A227550 A286384 A379010

Adjacent sequences: A194946 A194947 A194948 * A194950 A194951 A194952

Keyword

nonn,tabl

Author

Paul D. Hanna, Sep 05 2011

Status

approved