A195673 -id:A195673 - cp's OEIS frontend

A195662 Triangle T(n,k) read by rows: T(0,0)= -3, T(1,0)= 2, T(1,1) = 0 and T(n,k) = T(n-1,k) -T(n-2,k-2) otherwise.

-3, 2, 0, 2, 0, 3, 2, 0, 1, 0, 2, 0, -1, 0, -3, 2, 0, -3, 0, -4, 0, 2, 0, -5, 0, -3, 0, 3, 2, 0, -7, 0, 0, 0, 7, 0, 2, 0, -9, 0, 5, 0, 10, 0, -3, 2, 0, -11, 0, 12, 0, 10, 0, -10, 0, 2, 0, -13, 0, 21, 0, 5, 0, -20, 0, 3, 2, 0, -15, 0, 32, 0, -7, 0, -30, 0, 13, 0

Offset: 0

Paul Curtz, Sep 22 2011

In the notation of A195673, this defines polynomials P(n,x,p=-3,q=2), where p and q are the values of the constant and linear order for n=0 and 1.
Row sums -- the value P(n,1,-3,2) of the polynomial -- are A130848(n+5).
For general seed values in the two top rows of the triangle, the recurrence T(n,k) = T(n-1,k) - T(n-2,k-2) defines the triangle
p;
q,   0;
q,   0,      -p;
q,   0,    -p-q, 0;
q,   0,   -p-2q, 0,     p;
q,   0,   -p-3q, 0,  2p+q,  0;
and a companion triangle by adding 1 to both seed values:
p+1;
q+1, 0;
q+1, 0,    -p-1;
q+1, 0,  -p-q-2, 0;
q+1, 0, -p-2q-3, 0,    p+1;
q+1, 0, -p-3q-4, 0, 2p+q+3, 0;
The point-by-point difference between two companions is P(n,x,p+1,q+1) - P(n,x,p,q) = S(n,x) as represented (with increasing exponents) by A053119.
Examples of such triangles are A053119 (p=q=1), A192575 (p=1, q=2),
A162514 (p=2, q=1, up to a sign factor), A192011 (p=-1, q=2), A135929 (p=-2, q=1, apart from a irregular leading T(0,0)).

			The first few rows are
-3;
2, 0;
2, 0,   3;
2, 0,   1, 0;
2, 0,  -1, 0, -3;
2, 0,  -3, 0, -4, 0;
2, 0,  -5, 0, -3, 0,  3;
2, 0,  -7, 0,  0, 0,  7, 0;
2, 0,  -9, 0,  5, 0, 10, 0,  -3;
2, 0, -11, 0, 12, 0, 10, 0, -10, 0;
2, 0, -13, 0, 21, 0,  5, 0, -20, 0, 3;

Cf. A135929, A192011.

p = -3; q = 2; t[0, 0] = p; t[, 0] = q; t[, ?OddQ] = 0; t[n, k_] /; k > n = 0; t[n_ /; n >= 0, k_ /; k >= 0] := t[n, k] = t[n-1, k] - t[n-2, k-2]; Table[t[n, k], {n, 0, 11}, {k, 0, n}] // Flatten (* Jean-François Alcover, Nov 27 2012 *)

T(n,0) = 2 (n>0).
T(n,2) = -A060747(n-3), n>2.
T(n,4) = A028347(n-5), n>6.
T(2n,2n) = -3*(-1)^n ; T(n, 2k-1) = 0 ; T(2n+1,2n) = -(3n-2)*(-1)^n.  - M. F. Hasler, Sep 28 2011

cp's OEIS Frontend

A195662 Triangle T(n,k) read by rows: T(0,0)= -3, T(1,0)= 2, T(1,1) = 0 and T(n,k) = T(n-1,k) -T(n-2,k-2) otherwise.

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