A145152 - OEIS

%I #12 Dec 21 2014 16:23:53

%S 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,

%T -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,

%U 35,-34,15,8,-19,15,-6,1,0,8,-28,56,-69,49,-7,-27,34,-21,7,-1,0,9,-36,84

%N 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.

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

%H Alois P. Heinz, <a href="/A145152/b145152.txt">Rows n = -3..137, flattened</a>

%e Row k=1 gives [1, 0, 0, 1, 0], so A145153(1,5) = A145153(1,4) + A145153(1,1) = 2.

%e Triangle begins:

%e 0;

%e 0,  0;

%e 0,  0, 0;

%e 1,  0, 0, 1;

%e 1,  0, 0, 1,  0;

%e 2, -1, 0, 1, -1, 0;

%e 3, -3, 1, 1, -2, 1, 0;

%p 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);

%t 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 *)

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

%K sign,tabl

%O -3,16

%A _Alois P. Heinz_, Oct 03 2008