A193002 Triangle T(n,k)=0 (k odd), T(0,0)=-3, T(n,0)=1 (n > 0) and T(n,k) = T(n-1,k) - T(n-2,k-2).

-3, 1, 0, 1, 0, 3, 1, 0, 2, 0, 1, 0, 1, 0, -3, 1, 0, 0, 0, -5, 0, 1, 0, -1, 0, -6, 0, 3, 1, 0, -2, 0, -6, 0, 8, 0, 1, 0, -3, 0, -5, 0, 14, 0, -3, 1, 0, -4, 0, -3, 0, 20, 0, -11, 0, 1, 0, -5, 0, 0, 0, 25, 0, -25, 0, 3, 1, 0, -6

Offset: 0

Author

Paul Curtz, Jul 14 2011

Keywords

• sign
• easy
• tabl

Comments

Consider an array with recurrence BB(m,n) = BB(m,n-1) + BB(m-1,n), m >= 0:

3, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1,

3, 2, 1, 0, -1, -2, -3, -4, -5, -6, -7,

3, 5, 6, 6, 5, 3, 0, -4, -9, -15, -22,

3, 8, 14, 20, 25, 28, 28, 24, 15, 0, -22,

3, 11, 25, 45, 70, 98, 126, 150, 165, 165, 143,

3, 14, 39, 84, 154, 252, 378, 528, 693, 858, 1001,

with BB(m,n) = (3m-n)*binomial(n+m-1,n)/m if m > 0. So the BB are polynomials of degree m in n:

BB(1,n) = -(n-3)/1,

BB(2,n) = -(n-6)*(n+1)/2, (see A055999)

BB(3,n) = -(n-9)*(n+1)*(n+2)/6,

BB(4,n) = -(n-12)*(n+1)*(n+2)*(n+3)/24,

BB(5,n) = -(n-15)*(n+1)*(n+2)*(n+3)*(n+4)/120.

Columns in the array are A010701, A016789, A095794, A005564, A059302.

T(n,k) is a zero-padded, column-shifted, sign-modified transpose of this array.

Examples

			Triangle begins
  -3;
   1,   0;
   1,   0,   3;
   1,   0,   2,   0;
   1,   0,   1,   0,  -3;
   1,   0,   0,   0,  -5,   0;
   1,   0,  -1,   0,  -6,   0,   3;
   1,   0,  -2,   0,  -6,   0,   8,   0;
   1,   0,  -3,   0,  -5,   0,  14,   0,  -3;
   1,   0,  -4,   0,  -3,   0,  20,   0, -11,   0;
		

Crossrefs

Cf. A174559.

Programs

Maple

BB := proc(m,n) if m=0 then if n= 0 then 3 ; else -1; end if; else (3*m-n)*binomial(n+m-1,n)/m ; end if; end proc:
A193002 := proc(n,k) if type(k,'odd') then 0; else (-1)^(1+k/2)*BB(k/2,n-k) ; end if; end proc:
seq(seq(A193002(n,k),k=0..n),n=0..15) ; # R. J. Mathar, Aug 30 2011

Formula

Sum_{k=0..n} T(n,k) = A130806(n+5). (row sums)

Sum_{k=0..n} (-1)^(k/2)*T(n,k) = -A000032(n-2). (alternating row sums)

T(n,k) = (-1)^(1+k/2)*BB(k/2,n-k). - R. J. Mathar, Aug 30 2011

T(n,2k) = (-1)^(1+k)*(5-n/k)*binomial(n-k-1,k-1), k > 0. - R. J. Mathar, Aug 30 2011