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

1,2

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

Links

Alois P. Heinz, Rows n = 1..141, flattened

N. J. A. Sloane, Transforms

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.

Example

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;

Maple

T:= proc(n, k) option remember; `if`(n<k, `if`(n=0, 1, 0),
       add(binomial(n-k, j) *T(j, k), j=0..n-k))
    end:
seq(seq(T(n, k), k=1..n), n=1..14);

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 *)

Crossrefs

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

Cf. A007318.

Sequence in context: A206563 A299779 A323954 * A282988 A113767 A352566

Adjacent sequences: A143980 A143981 A143982 * A143984 A143985 A143986

Keyword

eigen,nonn,tabl

Author

Alois P. Heinz, Sep 06 2008

Status

approved