A304223 Triangle read by rows: T(0,0)=1; T(n,k) = T(n-1,k)-2*T(n-2,k-1)+2*T(n-3,k-2) for k = 0..floor(2*n/3); T(n,k)=0 for n or k < 0.

1, 1, 1, -2, 1, -4, 2, 1, -6, 8, 1, -8, 18, -8, 1, -10, 32, -32, 4, 1, -12, 50, -80, 36, 1, -14, 72, -160, 136, -24, 1, -16, 98, -280, 360, -160, 8, 1, -18, 128, -448, 780, -592, 128, 1, -20, 162, -672, 1484, -1632, 720, -64

Offset: 0

Author

Shara Lalo, May 08 2018

Keywords

• tabf
• easy
• sign

Comments

The numbers in rows of the triangle are along skew diagonals pointing top-left in center-justified triangle given in A304209.

The coefficients in the expansion of 1/(1-x+2*x^2-2*x^3) are given by the sequence generated by the row sums.

Examples

			Triangle begins:
  1;
  1;
  1,  -2;
  1,  -4,   2;
  1,  -6,   8;
  1,  -8,  18,   -8;
  1, -10,  32,  -32,    4;
  1, -12,  50,  -80,   36;
  1, -14,  72, -160,  136,   -24;
  1, -16,  98, -280,  360,  -160,    8;
  1, -18, 128, -448,  780,  -592,  128;
  1, -20, 162, -672, 1484, -1632,  720,  -64;
  1, -22, 200, -960, 2576, -3752, 2624, -640,  16;
  ...
		

References

• Shara Lalo and Zagros Lalo, Polynomial Expansion Theorems and Number Triangles, Zana Publishing, 2018, ISBN: 978-1-9995914-0-3, pp. 204, 205.

Links

• Shara Lalo, Left-justified Triangle

Crossrefs

Row sums is A077953.

Cf. A304209.

Programs

PARI

T(n,k) = if ((n<0) || (k<0), 0, if ((n==0) && (k==0), 1, T(n-1,k)-2*T(n-2,k-1)+2*T(n-3,k-2)));
tabf(nn) = for (n=0, nn, for (k=0, 2*n\3, print1(T(n,k), ", ")); print); \\ Michel Marcus, May 10 2018