%I #18 Sep 05 2018 02:27:14
%S 1,1,1,4,1,8,1,12,16,1,16,48,1,20,96,64,1,24,160,256,1,28,240,640,256,
%T 1,32,336,1280,1280,1,36,448,2240,3840,1024,1,40,576,3584,8960,6144,1,
%U 44,720,5376,17920,21504,4096
%N Triangle read by rows: T(0,0)= 1; T(n,k)= T(n-1,k) + 4*T(n-2,k-1) for k = 0..floor(n/2); T(n,k)=0 for n or k < 0.
%C The numbers in rows of the triangle are along skew diagonals pointing top-right in center-justified triangle given in A013611 ((1+4*x)^n).
%C The coefficients in the expansion of 1/(1-x-4*x^2) are given by the sequence generated by the row sums.
%C If s(n) is the row sum at n, then the ratio s(n)/s(n-1) is approximately 2.5615528128...: A222132 (sqrt(4 + sqrt(4 + sqrt(4 + sqrt(4 + ... ))))), 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, pp. 70, 72, 371, 372.
%H Shara Lalo, <a href="/A305834/a305834.pdf">Right justified triangle</a>
%H Shara Lalo, <a href="/A305834/a305834_1.pdf">Skew diagonals in triangle A013611</a>
%F G.f.: 1/(1 - t*x - 4*t^2).
%F Column k is binomial (n + k - 1, k) * 4^k.
%e Triangle begins:
%e 1;
%e 1;
%e 1, 4;
%e 1, 8;
%e 1, 12, 16;
%e 1, 16, 48;
%e 1, 20, 96, 64;
%e 1, 24, 160, 256;
%e 1, 28, 240, 640, 256;
%e 1, 32, 336, 1280, 1280;
%e 1, 36, 448, 2240, 3840, 1024;
%e 1, 40, 576, 3584, 8960, 6144;
%e 1, 44, 720, 5376, 17920, 21504, 4096;
%e 1, 48, 880, 7680, 32256, 57344, 28672;
%e 1, 52, 1056, 10560, 53760, 129024, 114688, 16384;
%e 1, 56, 1248, 14080, 84480, 258048, 344064, 131072;
%e 1, 60, 1456, 18304, 126720, 473088, 860160, 589824, 65536;
%e 1, 64, 1680, 23296, 183040, 811008, 1892352, 1966080, 589824;
%t t[0, 0] = 1; t[n_, k_] := If[n < 0 || k < 0, 0, t[n - 1, k] + 4 t[n - 2, k - 1]]; Table[t[n, k], {n, 0, 12}, {k, 0, Floor[n/2]}] // Flatten
%Y Row sums give A006131.
%Y Cf. A000012 (column 0), A008586 (column 1), A035008 (column 2), A141478 (column 3), A120054 (column 4).
%Y Cf. A013611.
%Y Cf. A222132.
%K tabf,nonn,easy
%O 0,4
%A _Shara Lalo_, Jun 11 2018
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