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A317498 Triangle read by rows of coefficients in expansions of (-2 + 3x)^n, where n is nonnegative integer. 4

%I

%S 1,-2,3,4,-12,9,-8,36,-54,27,16,-96,216,-216,81,-32,240,-720,1080,

%T -810,243,64,-576,2160,-4320,4860,-2916,729,-128,1344,-6048,15120,

%U -22680,20412,-10206,2187,256,-3072,16128,-48384,90720,-108864,81648,-34992,6561,-512,6912,-41472,145152,-326592,489888,-489888,314928,-118098,19683

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

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

%C This is a signed version of A013620.

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

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

%C 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.

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

%H Eric W. Weisstein, <a href="http://mathworld.wolfram.com/FermatPolynomial.html">Fermat Polynomial</a>.

%H Zagros Lalo, <a href="/A317498/a317498_1.pdf">First layer skew diagonals in center-justified triangle of coefficients in expansion of (-2 + 3 x)^n.</a>

%F 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.

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

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

%e Triangle begins:

%e 1;

%e -2, 3;

%e 4, -12, 9;

%e -8, 36, -54, 27;

%e 16, -96, 216, -216, 81;

%e -32, 240, -720, 1080, -810, 243;

%e 64, -576, 2160, -4320, 4860, -2916, 729;

%e -128, 1344, -6048, 15120, -22680, 20412, -10206, 2187;

%e 256, -3072, 16128, -48384, 90720, -108864, 81648, -34992, 6561;

%e -512, 6912, -41472, 145152, -326592, 489888, -489888, 314928, -118098, 19683;

%t 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 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.

%t Table[CoefficientList[(-2 + 3 x)^n, x], {n, 0, 9}] // Flatten.

%o (PARI) trianglerows(n) = my(v=[]); for(k=0, n-1, v=Vec((-2+3*x)^k + O(x^(k+1))); print(v))

%o /* Print initial 10 rows of triangle as follows */

%o trianglerows(10) \\ _Felix Fröhlich_, Jul 31 2018

%o (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

%Y Row sums give A000012.

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

%Y Cf. A302747, A303941.

%K tabf,sign,easy

%O 0,2

%A _Zagros Lalo_, Jul 31 2018

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Last modified March 8 01:41 EST 2021. Contains 341934 sequences. (Running on oeis4.)