A304213 Triangle read by rows: T(0,0) = 1; T(n,k) = 2*T(n-1,k) - 2*T(n-2,k-1) + T(n-3,k-2) for k = 0..floor(2*n/3); T(n,k)=0 for n or k < 0.

1, 2, 4, -2, 8, -8, 1, 16, -24, 8, 32, -64, 36, -4, 64, -160, 128, -32, 1, 128, -384, 400, -160, 18, 256, -896, 1152, -640, 136, -6, 512, -2048, 3136, -2240, 720, -80, 1, 1024, -4608, 8192, -7168, 3120, -592, 32, 2048, -10240, 20736, -21504, 11872, -3264, 360, -8, 4096, -22528, 51200, -61440, 41216, -15008, 2624, -160, 1

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-right in center-justified triangle given in A304209.

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

Examples

			Triangle begins:
      1;
      2;
      4,      -2;
      8,      -8,      1;
     16,     -24,      8;
     32,     -64,     36,      -4;
     64,    -160,    128,     -32,      1;
    128,    -384,    400,    -160,     18;
    256,    -896,   1152,    -640,    136,      -6;
    512,   -2048,   3136,   -2240,    720,     -80,     1;
   1024,   -4608,   8192,   -7168,   3120,    -592,    32;
   2048,  -10240,  20736,  -21504,  11872,   -3264,   360,     -8;
   4096,  -22528,  51200,  -61440,  41216,  -15008,  2624,   -160,   1;
   8192,  -49152, 123904, -168960, 133632,  -60928, 14896,  -1632,  50;
  16384, -106496, 294912, -450560, 410880, -225792, 71680, -11776, 780, -10;
  ...
		

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 similar to A021823.

Cf. A304209.

Programs

PARI

T(n,k) = if ((n<0) || (k<0), 0, if ((n==0) && (k==0), 1, 2*T(n-1,k)-2*T(n-2,k-1)+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