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A317498
Triangle read by rows of coefficients in expansions of (-2 + 3x)^n, where n is nonnegative integer.
4
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.
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).
Sequence in context: A096864 A373576 A013620 * A081837 A119799 A036779
KEYWORD
tabl,sign,easy
AUTHOR
Zagros Lalo, Jul 31 2018
STATUS
approved