A145152 Triangle T(k,m), k>= - 3, 1<=m<=k + 4, read by rows: T(k,m) is the coefficient of a_k(n - m) in the recursive evaluation of a_k(n), where a_k is the k - th column sequence of A145153.

0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 2, -1, 0, 1, -1, 0, 3, -3, 1, 1, -2, 1, 0, 4, -6, 4, 0, -3, 3, -1, 0, 5, -10, 10, -4, -3, 6, -4, 1, 0, 6, -15, 20, -14, 1, 9, -10, 5, -1, 0, 7, -21, 35, -34, 15, 8, -19, 15, -6, 1, 0, 8, -28, 56, -69, 49, -7, -27, 34, -21, 7, -1, 0, 9, -36, 84

Offset

-3,16

Comments

Rows -3<=k<0 are used to form the shape of a perfect triangle.

Links

Alois P. Heinz, Rows n = -3..137, flattened

Example

Row k=1 gives [1, 0, 0, 1, 0], so A145153(1,5) = A145153(1,4) + A145153(1,1) = 2.
Triangle begins:
0;
0,  0;
0,  0, 0;
1,  0, 0, 1;
1,  0, 0, 1,  0;
2, -1, 0, 1, -1, 0;
3, -3, 1, 1, -2, 1, 0;

Maple

cor:= proc(k) option remember; `if`(k<=0, [1, 0, 0, 1], [seq(coeff( -(1-x-x^4) *(1-x)^(k-1), x, j), j=1..k+3)]) end: T:= proc(k, m) local l, j; l:= cor(k); `if`(k<0 or m<0 or m>nops(l), 0, l[m]) end: seq(seq(T(k, m), m=1..k+4), k=-3..13);

Mathematica

cor[k_] := cor[k] = If[k <= 0, {1, 0, 0, 1}, Table[Coefficient[-(1-x-x^4)*(1-x)^(k-1), x, j], {j, 1, k+3}]]; T[k_, m_] := Module[{l, j}, l = cor[k]; If[k<0 || m<0 || m>Length[l], 0, l[[m]]]]; Table[Table[T[k, m], {m, 1, k+4}], {k, -3, 13}] // Flatten (* Jean-François Alcover, Jan 15 2014, translated from Maple *)

Crossrefs

Cf. A145153, A017898, A003269, A098578, A145131, A145132, A145133, A145134, A145135, A145136, A145137.

Sequence in context: A341685 A069844 A233006 * A124327 A346837 A082596

Adjacent sequences: A145149 A145150 A145151 * A145153 A145154 A145155

Keyword

sign,tabl

Author

Alois P. Heinz, Oct 03 2008

Status

approved