A132844 Triangle, read by rows, where T(n,k) = {T^[(n+k)/2]}( [(n+k)/2], k) for n>=k>=0, so that antidiagonal {2n} equals row n of T^n for n>=0 and odd antidiagonals equal even antidiagonals.

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

Offset

0,8

Comments

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

Links

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened

Formula

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

Example

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 4th 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; ...

Mathematica

t[n_, k_] := t[n, k] = Module[{q = Quotient[n+k, 2], m, p}, m = Table[ Which[c < r-1, t[r-1, c-1], c <= r, 1, True, 0], {r, 1, q+1}, {c, 1, q+1}]; p = MatrixPower[m, q]; If[k > q, 0, p[[q+1, k+1]]]]; Table[t[n, k], {n, 0, 11}, {k, 0, n}] // Flatten (* Jean-François Alcover, Oct 02 2013, after Pari *)

Prog

(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]}

Crossrefs

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

Sequence in context: A104467 A132463 A153901 * A006843 A324797 A049456

Adjacent sequences: A132841 A132842 A132843 * A132845 A132846 A132847

Keyword

nonn,tabl

Author

Paul D. Hanna, Sep 17 2007

Status

approved