A132845 Triangle, read by rows, where row n equals row n of matrix power A132844^n for n>=0, where triangle A132844 is defined by: A132844(n,k) = T( [(n+k)/2], k) for n>=k>=0.

1, 1, 1, 3, 2, 1, 13, 9, 3, 1, 73, 42, 18, 4, 1, 466, 270, 95, 30, 5, 1, 3309, 1785, 693, 179, 45, 6, 1, 25425, 13657, 4893, 1463, 301, 63, 7, 1, 209717, 108606, 40506, 11104, 2726, 468, 84, 8, 1, 1837168, 943677, 338277, 99177, 22239, 4653, 687, 108, 9, 1

Offset

0,4

Links

Table of n, a(n) for n=0..54.

Formula

T(n,k) = [A132844^n](n,k) where A132844(n,k) = T( [(n+k)/2], k) for n>=k>=0.

Example

Triangle T begins:
1;
1, 1;
3, 2, 1;
13, 9, 3, 1;
73, 42, 18, 4, 1;
466, 270, 95, 30, 5, 1;
3309, 1785, 693, 179, 45, 6, 1;
25425, 13657, 4893, 1463, 301, 63, 7, 1;
209717, 108606, 40506, 11104, 2726, 468, 84, 8, 1;
1837168, 943677, 338277, 99177, 22239, 4653, 687, 108, 9, 1;
16995545, 8534290, 3110310, 873440, 213415, 40707, 7440, 965, 135, 10, 1; ...
Triangle A132844 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; ...
where column k of A132844 equals column k of T with terms repeated;
then row n of T equals row n of A132844^n as illustrated below.
Matrix square A132844^2 begins:
1;
2, 1;
3, 2, 1; <-- row 2 of T
5, 5, 2, 1;
12, 9, 7, 2, 1;
25, 31, 13, 9, 2, 1; ...
Matrix cube A132844^3 begins:
1;
3, 1;
6, 3, 1;
13, 9, 3, 1; <-- row 3 of T
33, 22, 12, 3, 1;
87, 75, 31, 15, 3, 1; ...
Matrix 4th power A132844^4 begins:
1;
4, 1;
10, 4, 1;
26, 14, 4, 1;
73, 42, 18, 4, 1; <-- row 4 of T
220, 151, 58, 22, 4, 1; ...

Mathematica

t[n_, k_] := t[n, k] = Module[{m, p}, m = Table[ Which[c < r, t[Quotient[r+c, 2]-1, c-1], c == r, 1, True, 0], {r, 1, n+1}, {c, 1, n+1}]; p = MatrixPower[m, n]; If[k > n, 0, p[[n+1, k+1]]]]; Table[t[n, k], {n, 0, 9}, {k, 0, n}] // Flatten (* Jean-François Alcover, Oct 02 2013, after Pari *)

Prog

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

Crossrefs

Cf. A132844 (triangle); columns: A132846, A132847, A132848, A132849.

Sequence in context: A104980 A316566 A134090 * A129652 A154921 A127126

Adjacent sequences: A132842 A132843 A132844 * A132846 A132847 A132848

Keyword

nonn,tabl

Author

Paul D. Hanna, Sep 17 2007

Status

approved