This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A317498 #24 Aug 08 2025 06:40:58
%S A317498 1,-2,3,4,-12,9,-8,36,-54,27,16,-96,216,-216,81,-32,240,-720,1080,
%T A317498 -810,243,64,-576,2160,-4320,4860,-2916,729,-128,1344,-6048,15120,
%U A317498 -22680,20412,-10206,2187,256,-3072,16128,-48384,90720,-108864,81648,-34992,6561,-512,6912,-41472,145152,-326592,489888,-489888,314928,-118098,19683
%N A317498 Triangle read by rows of coefficients in expansions of (-2 + 3*x)^n, where n is nonnegative integer.
%C A317498 Row n gives coefficients in expansion of (-2 + 3*x)^n.
%C A317498 This is a signed version of A013620.
%C A317498 The coefficients in the expansion of 1/(1-x) are given by the sequence generated by the row sums.
%C A317498 The row sums give A000012 (The simplest sequence of positive numbers: the all 1's sequence).
%C A317498 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 + 3*x)^n, see links.
%D A317498 Shara Lalo and Zagros Lalo, Polynomial Expansion Theorems and Number Triangles, Zana Publishing, 2018, ISBN: 978-1-9995914-0-3, Pages 394-396.
%H A317498 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>
%H A317498 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/FermatPolynomial.html">Fermat Polynomial</a>.
%F A317498 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 A317498 T(n, k) = ((-2)^(n - k) 3^k)/((n - k)! k!) n! for k = 0,1..n.
%F A317498 G.f.: 1 / (1 + 2*x - 3*x*t).
%e A317498 Triangle begins:
%e A317498 1;
%e A317498 -2, 3;
%e A317498 4, -12, 9;
%e A317498 -8, 36, -54, 27;
%e A317498 16, -96, 216, -216, 81;
%e A317498 -32, 240, -720, 1080, -810, 243;
%e A317498 64, -576, 2160, -4320, 4860, -2916, 729;
%e A317498 -128, 1344, -6048, 15120, -22680, 20412, -10206, 2187;
%e A317498 256, -3072, 16128, -48384, 90720, -108864, 81648, -34992, 6561;
%e A317498 -512, 6912, -41472, 145152, -326592, 489888, -489888, 314928, -118098, 19683;
%e A317498 ...
%t A317498 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 A317498 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 A317498 Table[CoefficientList[(-2 + 3 x)^n, x], {n, 0, 9}] // Flatten
%o A317498 (PARI) trianglerows(n) = my(v=[]); for(k=0, n-1, v=Vec((-2+3*x)^k + O(x^(k+1))); print(v))
%o A317498 /* Print initial 10 rows of triangle as follows */
%o A317498 trianglerows(10) \\ _Felix Fröhlich_, Jul 31 2018
%o A317498 (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 A317498 Row sums give A000012.
%Y A317498 Cf. A013620 ((2+3*x)^n).
%Y A317498 Cf. A302747, A303941.
%K A317498 tabl,sign,easy
%O A317498 0,2
%A A317498 _Zagros Lalo_, Jul 31 2018