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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. 6
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 (list; table; graph; refs; listen; history; text; internal format)
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

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Last modified December 9 02:44 EST 2021. Contains 349625 sequences. (Running on oeis4.)