A317498 Triangle read by rows of coefficients in expansions of (-2 + 3x)^n, where n is nonnegative integer.

1, -2, 3, 4, -12, 9, -8, 36, -54, 27, 16, -96, 216, -216, 81, -32, 240, -720, 1080, -810, 243, 64, -576, 2160, -4320, 4860, -2916, 729, -128, 1344, -6048, 15120, -22680, 20412, -10206, 2187, 256, -3072, 16128, -48384, 90720, -108864, 81648, -34992, 6561, -512, 6912, -41472, 145152, -326592, 489888, -489888, 314928, -118098, 19683

Offset

0,2

Comments

Row n gives coefficients in expansion of (-2 + 3 x)^n.

This is a signed version of A013620.

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

The row sums give A000012 (The simplest sequence of positive numbers: the all 1's sequence).

The numbers in rows of triangles in A302747 and A303941 (Triangle of coefficients of Fermat polynomials) are along first layer skew diagonals pointing top-right and top-left in center-justified triangle of coefficients in expansions of (-2 + 3x)^n, see links.

References

Shara Lalo and Zagros Lalo, Polynomial Expansion Theorems and Number Triangles, Zana Publishing, 2018, ISBN: 978-1-9995914-0-3, Pages 394-396.

Links

Table of n, a(n) for n=0..54.

Zagros Lalo, First layer skew diagonals in center-justified triangle of coefficients in expansion of (-2 + 3 x)^n.

Eric Weisstein's World of Mathematics, Fermat Polynomial.

Formula

T(0,0) = 1; T(n,k) = -2 * T(n-1,k) + 3 * T(n-1,k-1) for k = 0,1,...,n and T(n,k)=0 for n or k < 0.

T(n, k) = ((-2)^(n - k) 3^k)/((n - k)! k!) n! for k = 0,1..n.

Has the g.f.: 1 / (1 + 2x - 3x t).

Example

Triangle begins:
     1;
    -2,     3;
     4,   -12,      9;
    -8,    36,    -54,     27;
    16,   -96,    216,   -216,      81;
   -32,   240,   -720,   1080,    -810,     243;
    64,  -576,   2160,  -4320,    4860,   -2916,     729;
  -128,  1344,  -6048,  15120,  -22680,   20412,  -10206,   2187;
   256, -3072,  16128, -48384,   90720, -108864,   81648, -34992,    6561;
  -512,  6912, -41472, 145152, -326592,  489888, -489888, 314928, -118098, 19683;
  ...

Mathematica

t[0, 0] = 1; t[n_, k_] :=  t[n, k] =   If[n < 0 || k < 0, 0, -2  t[n - 1, k] + 3  t[n - 1, k - 1]]; Table[t[n, k], {n, 0, 9}, {k, 0, n}] // Flatten.
t[n_, k_] := t[n, k] = ((-2)^(n - k) 3^k)/((n - k)! k!) n!; Table[t[n, k], {n, 0, 9}, {k, 0, n} ]  // Flatten.
Table[CoefficientList[(-2 + 3 x)^n, x], {n, 0, 9}] // Flatten.

Prog

(PARI) trianglerows(n) = my(v=[]); for(k=0, n-1, v=Vec((-2+3*x)^k + O(x^(k+1))); print(v))
/* Print initial 10 rows of triangle as follows */
trianglerows(10) \\ Felix Fröhlich, Jul 31 2018
(GAP) Flat(List([0..8], n->List([0..n], k->(-2)^(n-k)*3^k/(Factorial(n-k)*Factorial(k))*Factorial(n)))); # Muniru A Asiru, Aug 01 2018

Crossrefs

Row sums give A000012.

Cf. A013620 ((2+3x)^n).

Cf. A302747, A303941.

Sequence in context: A345276 A096864 A013620 * A119799 A036779 A037339

Adjacent sequences: A317495 A317496 A317497 * A317499 A317500 A317501

Keyword

tabl,sign,easy

Author

Zagros Lalo, Jul 31 2018

Status

approved