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A303901 Triangle read by rows of coefficients in expansion of (3-2x)^n, where n is a nonnegative integer.

%I A303901 #39 Mar 17 2025 02:39:52
%S A303901 1,3,-2,9,-12,4,27,-54,36,-8,81,-216,216,-96,16,243,-810,1080,-720,
%T A303901 240,-32,729,-2916,4860,-4320,2160,-576,64,2187,-10206,20412,-22680,
%U A303901 15120,-6048,1344,-128,6561,-34992,81648,-108864,90720,-48384,16128,-3072,256,19683,-118098,314928,-489888,489888,-326592,145152,-41472,6912,-512
%N A303901 Triangle read by rows of coefficients in expansion of (3-2x)^n, where n is a nonnegative integer.
%C A303901 This is a signed version of A038220.
%C A303901 Row n gives coefficients in expansion of (3-2x)^n.
%C A303901 The numbers in rows of triangles in A302747 and A303941 are along skew diagonals pointing top-left and top-right in center-justified triangle of coefficients in expansions of (3-2x)^n (A303901).
%C A303901 This is the lower triangular Riordan matrix (1/(1-3*t), -2*t/(1-3*t)), hence a convolution matrix. See the g.f.s. - _Wolfdieter Lang_, Jun 28 2018
%D A303901 Shara Lalo and Zagros Lalo, Polynomial Expansion Theorems and Number Triangles, Zana Publishing, 2018, ISBN: 978-1-9995914-0-3, pp. 394, 396, 398.
%H A303901 Zagros Lalo, <a href="/A303901/a303901_1.pdf">Center-justified Triangle</a>
%H A303901 Zagros Lalo, <a href="/A303901/a303901_2.pdf">Skew Diagonals in center-justified Triangle</a>
%F A303901 T(0,0) = 1; T(n,k) = 3*T(n-1,k) - 2*T(n-1,k-1) for k = 0,1,...,n; T(n,k)=0 for n or k < 0.
%F A303901 G.f. of row polynomials: 1 / (1 - 3*t + 2*t*x).
%F A303901 G.f. of column k: (-2*x)^k/(1-3*x)^(k+1), for k >= 0.
%e A303901 Triangle begins:
%e A303901   n \k 0     1       2       3      4      5       6      7      8    9  ...
%e A303901   --------------------------------------------------------------------------
%e A303901   0 |  1
%e A303901   1 |  3     -2
%e A303901   2 |  9     -12     4
%e A303901   3 |  27    -54     36     -8
%e A303901   4 |  81    -216    216    -96     16
%e A303901   5 |  243   -810    1080   -720    240    -32
%e A303901   6 |  729   -2916   4860   -4320   2160   -576    64
%e A303901   7 |  2187  -10206  20412  -22680  15120  -6048   1344   -128
%e A303901   8 |  6561  -34992  81648  -108864 90720  -48384  16128  -3072  256
%e A303901   9 |  19683 -118098 314928 -489888 489888 -326592 145152 -41472 6912 -512
%t A303901 For[i = 0, i < 4, i++, Print[CoefficientList[Expand[(3 - 2 x)^i],x]]]
%Y A303901 Cf. A013620 (unsigned), A000012 (row sums), A000351 (alternating row sums).
%K A303901 tabl,easy,sign
%O A303901 0,2
%A A303901 _Zagros Lalo_, May 02 2018
%E A303901 Edited - _Wolfdieter Lang_, Jun 28 2018