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A305834 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.

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