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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. 7
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

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.

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

Table of n, a(n) for n=0..48.

Zagros Lalo, Left-justified triangle

Eric W. Weisstein, Fermat Polynomial

EXAMPLE

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

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

PROG

(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

CROSSREFS

Row sums give A000225.

Some row sums give A001348.

Cf. A303901.

Sequence in context: A140985 A286676 A246379 * A176885 A257731 A257733

Adjacent sequences:  A303938 A303939 A303940 * A303942 A303943 A303944

KEYWORD

tabf,easy,sign

AUTHOR

Zagros Lalo, May 03 2018

STATUS

approved

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Last modified March 1 03:09 EST 2021. Contains 341732 sequences. (Running on oeis4.)