A303941 Triangle read by rows: T(0,0) = 1; T(n,k) = 3*T(n-1,k) - 2*T(n-2,k-1) for k = 0..floor(n/2); T(n,k)=0 for n or k < 0. Triangle of coefficients of Fermat polynomials.

1, 3, 9, -2, 27, -12, 81, -54, 4, 243, -216, 36, 729, -810, 216, -8, 2187, -2916, 1080, -96, 6561, -10206, 4860, -720, 16, 19683, -34992, 20412, -4320, 240, 59049, -118098, 81648, -22680, 2160, -32, 177147, -393660, 314928, -108864, 15120, -576, 531441, -1299078, 1180980, -489888, 90720, -6048, 64

Offset: 0

Author

Zagros Lalo, May 03 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 A303901 ((3-2x)^n).

Row n gives coefficients of Fermat polynomial.

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

Examples

			Triangle begins:
n\k |       0         1        2         3        4        5      6     7
----+--------------------------------------------------------------------
   0|       1
   1|       3
   2|       9        -2
   3|      27       -12
   4|      81       -54        4
   5|     243      -216       36
   6|     729      -810      216        -8
   7|    2187     -2916     1080       -96
   8|    6561    -10206     4860      -720       16
   9|   19683    -34992    20412     -4320      240
  10|   59049   -118098    81648    -22680     2160      -32
  11|  177147   -393660   314928   -108864    15120     -576
  12|  531441  -1299078  1180980   -489888    90720    -6048     64
  13| 1594323  -4251528  4330260  -2099520   489888   -48384   1344
  14| 4782969 -13817466 15588936  -8660520  2449440  -326592  16128  -128
  15|14348907 -44641044 55269864 -34642080 11547360 -1959552 145152 -3072
		

References

• Shara Lalo and Zagros Lalo, Polynomial Expansion Theorems and Number Triangles, Zana Publishing, 2018, ISBN: 978-1-9995914-0-3, pp. 70, 104, 394, 395.

Links

• Zagros Lalo, Left-justified triangle
• Eric Weisstein's World of Mathematics, Fermat Polynomial.

Crossrefs

Row sums give A000225.

Some row sums give A001348.

Cf. A303901.

Programs

Mathematica

t[0, 0] = 1; t[n_, k_] := If[n < 0 || k < 0, 0, 3 t[n - 1, k] - 2 t[n - 2, k - 1]]; Table[t[n, k], {n, 0, 14}, {k, 0, Floor[n/2]}] // Flatten

PARI

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