A303941 - OEIS

%I #53 Jan 26 2024 16:02:15

%S 1,3,9,-2,27,-12,81,-54,4,243,-216,36,729,-810,216,-8,2187,-2916,1080,

%T -96,6561,-10206,4860,-720,16,19683,-34992,20412,-4320,240,59049,

%U -118098,81648,-22680,2160,-32,177147,-393660,314928,-108864,15120,-576,531441,-1299078,1180980,-489888,90720,-6048,64

%N 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.

%C The numbers in rows of the triangle are along skew diagonals pointing top-right in center-justified triangle given in A303901 ((3-2x)^n).

%C Row n gives coefficients of Fermat polynomial.

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

%D 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.

%H Zagros Lalo, <a href="/A303941/a303941.pdf">Left-justified triangle</a>

%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/FermatPolynomial.html">Fermat Polynomial</a>.

%e Triangle begins:

%e n\k |       0         1        2         3        4        5      6     7

%e ----+--------------------------------------------------------------------

%e    0|       1

%e    1|       3

%e    2|       9        -2

%e    3|      27       -12

%e    4|      81       -54        4

%e    5|     243      -216       36

%e    6|     729      -810      216        -8

%e    7|    2187     -2916     1080       -96

%e    8|    6561    -10206     4860      -720       16

%e    9|   19683    -34992    20412     -4320      240

%e   10|   59049   -118098    81648    -22680     2160      -32

%e   11|  177147   -393660   314928   -108864    15120     -576

%e   12|  531441  -1299078  1180980   -489888    90720    -6048     64

%e   13| 1594323  -4251528  4330260  -2099520   489888   -48384   1344

%e   14| 4782969 -13817466 15588936  -8660520  2449440  -326592  16128  -128

%e   15|14348907 -44641044 55269864 -34642080 11547360 -1959552 145152 -3072

%t 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

%o (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)));

%o tabf(nn) = for (n=0, nn, for (k=0, n\2, print1(T(n,k), ", ")); print); \\ _Michel Marcus_, May 10 2018

%Y Row sums give A000225.

%Y Some row sums give A001348.

%Y Cf. A303901.

%K tabf,easy,sign

%O 0,2

%A _Zagros Lalo_, May 03 2018