0,8

Column k of triangle A132845 is equal to column k of this triangle but without repetition of terms.

T(n,k) = A132845( [(n+k)/2], k) where A132845(n,k) = [T^n](n,k) for n>=k>=0.

Triangle T begins:

1;

1, 1;

1, 1, 1;

1, 2, 1, 1;

3, 2, 3, 1, 1;

3, 9, 3, 4, 1, 1;

13, 9, 18, 4, 5, 1, 1;

13, 42, 18, 30, 5, 6, 1, 1;

73, 42, 95, 30, 45, 6, 7, 1, 1;

73, 270, 95, 179, 45, 63, 7, 8, 1, 1;

466, 270, 693, 179, 301, 63, 84, 8, 9, 1, 1;

466, 1785, 693, 1463, 301, 468, 84, 108, 9, 10, 1, 1;

3309, 1785, 4893, 1463, 2726, 468, 687, 108, 135, 10, 11, 1, 1; ...

Matrix square T^2 begins:

1;

2, 1;

3, 2, 1; <-- antidiagonals 4, 5, of T

5, 5, 2, 1;

12, 9, 7, 2, 1;

25, 31, 13, 9, 2, 1;

75, 63, 58, 17, 11, 2, 1; ...

Matrix cube T^3 begins:

1;

3, 1;

6, 3, 1;

13, 9, 3, 1; <-- antidiagonals 6, 7, of T

33, 22, 12, 3, 1;

87, 75, 31, 15, 3, 1;

265, 204, 132, 40, 18, 3, 1; ...

Matrix 4-th power T^4 begins:

1;

4, 1;

10, 4, 1;

26, 14, 4, 1;

73, 42, 18, 4, 1; <-- antidiagonals 8, 9, of T

220, 151, 58, 22, 4, 1;

717, 488, 253, 74, 26, 4, 1; ...

(PARI) {T(n, k)=local(M=matrix((n+k)\2+1, (n+k)\2+1, r, c, if(r>=c, if(r<=c+1, 1, T(r-1, c-1))))); (M^((n+k)\2))[(n+k)\2+1, k+1]}

Cf. A132845 (triangle); columns: A132846, A132847, A132848, A132849.

Adjacent sequences: A132841 A132842 A132843 this_sequence A132845 A132846 A132847

Sequence in context: A132462 A104467 A132463 this_sequence A006843 A049456 A117506

nonn,tabl

Paul D. Hanna (pauldhanna(AT)juno.com), Sep 17 2007