A065432 Triangle related to Catalan triangle: recurrence related to A033877 (Schroeder numbers).

1, 1, -1, 1, -2, 0, 1, -3, 1, 1, 1, -4, 3, 2, 0, 1, -5, 6, 2, -2, -2, 1, -6, 10, 0, -6, -4, 0, 1, -7, 15, -5, -11, -3, 5, 5, 1, -8, 21, -14, -15, 4, 15, 10, 0, 1, -9, 28, -28, -15, 19, 26, 6, -14, -14, 1, -10, 36, -48, -7, 42, 30, -16, -42, -28, 0, 1, -11, 45, -75, 14, 70, 16, -60, -70, -14, 42, 42, 1, -12, 55, -110, 54, 96, -28, -120, -70, 56, 126, 84, 0

Offset: 0

Author

Wouter Meeussen, Nov 16 2001

Keywords

• sign
• tabl

Comments

Sums of odd rows are 0, of even rows are the Catalan numbers (A000108) with alternating signs. Row sums of unsigned version give A065441.

Examples

			Triangle starts:
[0] 1;
[1] 1, -1;
[2] 1, -2,  0;
[3] 1, -3,  1,   1;
[4] 1, -4,  3,   2,   0;
[5] 1, -5,  6,   2,  -2, -2;
[6] 1, -6, 10,   0,  -6, -4,  0;
[7] 1, -7, 15,  -5, -11, -3,  5,  5;
[8] 1, -8, 21, -14, -15,  4, 15, 10,   0;
[9] 1, -9, 28, -28, -15, 19, 26,  6, -14, -14.
		

Programs

Mathematica

a[0, 0] := 1; a[n_, k_] := 0/;(k > n||n < 0||k < 0); a[n_, k_] := a[n, k] = a[n, k-1]-2a[n-1, k-1]+a[n-1, k]; Table[a[n, k], {n, 0, 16}, {k, 0, n}]

Extensions

More terms from Sean A. Irvine, Sep 03 2023