A317505 - OEIS

%I #8 Aug 23 2018 16:30:39

%S 1,-1,1,-1,2,1,-4,-1,6,1,-8,4,-1,10,-12,1,-12,24,-1,14,-40,8,1,-16,60,

%T -32,-1,18,-84,80,1,-20,112,-160,16,-1,22,-144,280,-80,1,-24,180,-448,

%U 240,-1,26,-220,672,-560,32,1,-28,264,-960,1120,-192,-1,30,-312,1320,-2016,672,1,-32,364,-1760,3360,-1792,64,-1,34,-420,2288,-5280,4032,-448

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

%C The numbers in rows of the triangle are along "second layer" skew diagonals pointing top-left in center-justified triangle given in A065109 ((2-x)^n) and along "second layer" skew diagonals pointing top-right in center-justified triangle given in A303872 ((-1+2x)^n), see links. (Note: First layer skew diagonals in center-justified triangles of coefficients in expansions of (2-x)^n and (-1+2x)^n are given in A133156 (coefficients of Chebyshev polynomials of the second kind) and A305098 respectively.) The coefficients in the expansion of 1/(1+x+2x^3) are given by the sequence generated by the row sums (see A077973).

%D Shara Lalo and Zagros Lalo, Polynomial Expansion Theorems and Number Triangles, Zana Publishing, 2018, ISBN: 978-1-9995914-0-3, pp. 139-141, 391-393.

%H Shara Lalo, <a href="/A317505/a317505.pdf">Second layer skew diagonals in center-justified triangle of coefficients in expansion of (2 - x)^n</a>

%H Shara Lalo, <a href="/A317505/a317505_1.pdf">Second layer skew diagonals in center-justified triangle of coefficients in expansion of (-1 + 2x)^n</a>

%F T(n,k) = (-1)^(n - 3k) * 2^k / ((n - 3k)! k!) * (n - 2k)! where n is a nonnegative integer and k = 0..floor(n/3).

%e Triangle begins:

%e    1;

%e   -1;

%e    1;

%e   -1,   2;

%e    1,  -4;

%e   -1,   6;

%e    1,  -8,    4;

%e   -1,  10,  -12;

%e    1, -12,   24;

%e   -1,  14,  -40,     8;

%e    1, -16,   60,   -32;

%e   -1,  18,  -84,    80;

%e    1, -20,  112,  -160,    16;

%e   -1,  22, -144,   280,   -80;

%e    1, -24,  180,  -448,   240;

%e   -1,  26, -220,   672,  -560,    32;

%e    1, -28,  264,  -960,  1120,  -192;

%e   -1,  30, -312,  1320, -2016,   672;

%e    1, -32,  364, -1760,  3360, -1792,   64;

%e   -1,  34, -420,  2288, -5280,  4032, -448;

%t t[n_, k_] := t[n, k] = (-1)^(n - 3k) * 2^k/((n - 3 k)! k!) * (n - 2 k)!; Table[t[n, k], {n, 0, 19}, {k, 0, Floor[n/3]} ]  // Flatten

%t t[0, 0] = 1; t[n_, k_] := t[n, k] = If[n < 0 || k < 0, 0, - t[n - 1, k] + 2 t[n - 3, k - 1]]; Table[t[n, k], {n, 0, 19}, {k, 0, Floor[n/3]}] // Flatten

%Y Row sums give A077973.

%Y Cf. A065109, A303872.

%Y Cf. A133156, A305098.

%K tabf,sign,easy

%O 0,5

%A _Shara Lalo_, Aug 02 2018