A143983 Triangle T(n,k), n>=1, 1<=k<=n, read by rows, where sequence a_k of column k has a_k(0)=1, followed by (k-1)-fold 0 and a_k(n) shifts k places down under binomial transform.

1, 2, 1, 5, 1, 1, 15, 2, 1, 1, 52, 5, 1, 1, 1, 203, 13, 2, 1, 1, 1, 877, 36, 6, 1, 1, 1, 1, 4140, 109, 17, 2, 1, 1, 1, 1, 21147, 359, 44, 7, 1, 1, 1, 1, 1, 115975, 1266, 112, 23, 2, 1, 1, 1, 1, 1, 678570, 4731, 304, 65, 8, 1, 1, 1, 1, 1, 1, 4213597, 18657, 918, 165, 30, 2, 1, 1, 1, 1, 1, 1

Offset: 1

Author

Alois P. Heinz, Sep 06 2008

Keywords

• eigen
• nonn
• tabl

Comments

The matrix inverse starts:

1;

-2, 1;

-3, -1, 1;

-8, -1, -1, 1;

-31, -3, 0, -1, 1;

-132, -7, -1, 0, -1, 1;

-616, -19, -4, 0, 0, -1, 1; - R. J. Mathar, Mar 22 2013

Examples

			T(5,2) = 5, because [1,3,3,1] * [1,0,1,1] = 5.
Triangle begins:
:      1;
:      2,    1;
:      5,    1,   1;
:     15,    2,   1,  1;
:     52,    5,   1,  1, 1;
:    203,   13,   2,  1, 1, 1;
:    877,   36,   6,  1, 1, 1, 1;
:   4140,  109,  17,  2, 1, 1, 1, 1;
:  21147,  359,  44,  7, 1, 1, 1, 1, 1;
: 115975, 1266, 112, 23, 2, 1, 1, 1, 1, 1;
		

Links

• Alois P. Heinz, Rows n = 1..141, flattened
• N. J. A. Sloane, Transforms

Crossrefs

Columns 1-6 give: A000110, A000994, A000996, A010748, A010749, A010750.

Cf. A007318.

Programs

Maple

T:= proc(n, k) option remember; `if`(n
				

Mathematica

t[n_, k_] := t[n, k] = If[n < k, If[n == 0, 1, 0], Sum[Binomial[n-k, j]*t[j, k], {j, 0, n-k}]]; Table[Table[t[n, k], {k, 1, n}], {n, 1, 13}] // Flatten (* Jean-François Alcover, Dec 18 2013, translated from Maple *)

Formula

T(n,k) = Sum_{j=0..n-k} C(n-k,j)*T(j,k) if n>=k, else T(n,k) = 1 if n=1, else T(n,k) = 0.