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 A318686 Triangle read by rows: T(0,0) = 1; T(n,k) = T(n-1,k) - 2*T(n-1,k-2) + T(n-1,k-3) for k = 0..3n; T(n,k)=0 for n or k < 0. 1

%I

%S 1,1,0,-2,1,1,0,-4,2,4,-4,1,1,0,-6,3,12,-12,-5,12,-6,1,1,0,-8,4,24,

%T -24,-26,48,-8,-28,24,-8,1,1,0,-10,5,40,-40,-70,120,20,-150,88,40,-75,

%U 40,-10,1,1,0,-12,6,60,-60,-145,240,120,-460,168,360,-401,48,180,-154,60,-12,1,1,0,-14,7,84,-84,-259,420,350,-1085,168,1400,-1197,-504,1342,-651,-252,476,-273,84,-14,1

%N Triangle read by rows: T(0,0) = 1; T(n,k) = T(n-1,k) - 2*T(n-1,k-2) + T(n-1,k-3) for k = 0..3n; T(n,k)=0 for n or k < 0.

%C Row n gives coefficients in expansion of (1 - 2*x^2 + x^3)^n. Row sum s(n) = 1 when n = 0 and s(n) = 0 when n > 0, see link. In the center-justified triangle, the sum of numbers along "first layer" skew diagonals pointing top-right are the coefficients in expansion of 1/(1 - x + 2 x^3 - x^4) and the sum of numbers along "first layer" skew diagonals pointing top-left are the coefficients in expansion of 1/(1 - x + 2*x^2 - x^4), see links.

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

%H Shara Lalo, <a href="/A318686/a318686.pdf">Centre-justified triangle of coefficients in expansions of (1 - 2 x^2 + x^3)^n</a>

%H Shara Lalo, <a href="/A318686/a318686_1.pdf">First layer skew diagonals in center-justified triangle of coefficients in expansion of (1 - 2 x^2 + x^3)^n</a>

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

%F G.f.: 1/(1 - t + 2*t x^2 - t*x^3).

%e Triangle begins:

%e 1;

%e 1, 0, -2, 1;

%e 1, 0, -4, 2, 4, -4, 1;

%e 1, 0, -6, 3, 12, -12, -5, 12, -6, 1;

%e 1, 0, -8, 4, 24, -24, -26, 48, -8, -28, 24, -8, 1;

%e 1, 0, -10, 5, 40, -40, -70, 120, 20, -150, 88, 40, -75, 40, -10, 1;

%e 1, 0, -12, 6, 60, -60, -145, 240, 120, -460, 168, 360, -401, 48, 180, -154, 60, -12, 1;

%e ...

%t t[0, 0] = 1; t[n_, k_] := t[n, k] = If[n < 0 || k < 0, 0, t[n - 1, k] - 2 t[n - 1, k - 2] + t[n - 1, k - 3]]; Table[t[n, k], {n, 0, 7}, {k, 0, 3 n} ] // Flatten

%K tabf,sign,easy

%O 0,4

%A _Shara Lalo_, Sep 06 2018

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Last modified December 1 01:37 EST 2021. Contains 349426 sequences. (Running on oeis4.)