A317506 - OEIS

A317506

Triangle read by rows: T(0,0) = 1; T(n,k) = 2 T(n-1,k) - T(n-4,k-1) for 0 <= k <= floor(n/4); T(n,k)=0 for n or k < 0.

%I #16 Jul 21 2021 22:50:06

%S 1,2,4,8,16,-1,32,-4,64,-12,128,-32,256,-80,1,512,-192,6,1024,-448,24,

%T 2048,-1024,80,4096,-2304,240,-1,8192,-5120,672,-8,16384,-11264,1792,

%U -40,32768,-24576,4608,-160,65536,-53248,11520,-560,1,131072,-114688,28160,-1792,10

%N Triangle read by rows: T(0,0) = 1; T(n,k) = 2 T(n-1,k) - T(n-4,k-1) for 0 <= k <= floor(n/4); T(n,k)=0 for n or k < 0.

%C The numbers in rows of the triangle are along "third layer" skew diagonals pointing top-right in center-justified triangle given in A065109 ((2-x)^n) and along "third layer" skew diagonals pointing top-left 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/(1-2x+x^4) are given by the sequence generated by the row sums. The row sums give A008937. If s(n) is the row sum at n, then the ratio s(n)/s(n-1) is approximately 1.83928675521416113... (A058265: Decimal expansion of the tribonacci constant t, the real root of x^3-x^2-x-1), when n approaches infinity.

%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="/A317506/a317506.pdf">Third layer skew diagonals in center-justified triangle of coefficients in expansion of (2 - x)^n</a>

%H Shara Lalo, <a href="/A317506/a317506_1.pdf">Third layer skew diagonals in center-justified triangle of coefficients in expansion of (-1 + 2x)^n</a>

%F T(n,k) = 2^(n - 4*k) * (-1)^k / ((n - 4*k)! k!) * (n - 3*k)! where n >= 0 and 0 <= k <= floor(n/4).

%e Triangle begins:

%e 1;

%e 2;

%e 4;

%e 8;

%e 16, -1;

%e 32, -4;

%e 64, -12;

%e 128, -32;

%e 256, -80, 1;

%e 512, -192, 6;

%e 1024, -448, 24;

%e 2048, -1024, 80;

%e 4096, -2304, 240, -1;

%e 8192, -5120, 672, -8;

%e 16384, -11264, 1792, -40;

%e 32768, -24576, 4608, -160;

%e 65536, -53248, 11520, -560, 1;

%e 131072, -114688, 28160, -1792, 10;

%e 262144, -245760, 67584, -5376, 60;

%t t[n_, k_] := t[n, k] = 2^(n - 4 k) * (-1)^k/((n - 4 k)! k!) * (n - 3 k)!; Table[t[n, k], {n, 0, 16}, {k, 0, Floor[n/4]} ] // Flatten

%t t[0, 0] = 1; t[n_, k_] := t[n, k] = If[n < 0 || k < 0, 0, 2 * t[n - 1, k] - t[n - 4, k - 1]]; Table[t[n, k], {n, 0, 16}, {k, 0, Floor[n/4]}] // Flatten

%Y Row sums give A008937.

%Y Cf. A065109, A303872.

%Y Cf. A133156, A305098.

%Y Cf. A058265.

%K tabf,sign,easy

%O 0,2

%A _Shara Lalo_, Aug 31 2018