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A317495 Triangle read by rows: T(0,0) = 1; T(n,k) =2 * T(n-1,k) + T(n-3,k-1) for k = 0..floor(n/3); T(n,k)=0 for n or k < 0.

%I A317495 #25 Sep 08 2022 08:46:22
%S A317495 1,2,4,8,1,16,4,32,12,64,32,1,128,80,6,256,192,24,512,448,80,1,1024,
%T A317495 1024,240,8,2048,2304,672,40,4096,5120,1792,160,1,8192,11264,4608,560,
%U A317495 10,16384,24576,11520,1792,60,32768,53248,28160,5376,280,1,65536,114688,67584,15360,1120,12
%N A317495 Triangle read by rows: T(0,0) = 1; T(n,k) =2 * T(n-1,k) + T(n-3,k-1) for k = 0..floor(n/3); T(n,k)=0 for n or k < 0.
%C A317495 The numbers in rows of the triangle are along a "second layer" of skew diagonals pointing top-left in center-justified triangle given in A013609 ((1+2*x)^n) and along a "second layer" of skew diagonals pointing top-right in center-justified triangle given in A038207 ((2+x)^n), see links. (Note: First layer skew diagonals in center-justified triangles of coefficients in expansions of (1+2*x)^n and (2+x)^n are given in A128099 and A207538 respectively.)
%C A317495 The coefficients in the expansion of 1/(1-2x-x^3) are given by the sequence generated by the row sums.
%C A317495 The row sums give A008998 and Pisot sequences E(4,9), P(4,9) when n > 1, see A020708.
%C A317495 If s(n) is the row sum at n, then the ratio s(n)/s(n-1) is approximately 2.205569430400..., when n approaches infinity.
%D A317495 Shara Lalo and Zagros Lalo, Polynomial Expansion Theorems and Number Triangles, Zana Publishing, 2018, ISBN: 978-1-9995914-0-3, pp. 358, 359.
%H A317495 Zagros Lalo, <a href="/A317495/a317495.pdf">Second layer of skew diagonals in center-justified triangle of coefficients in expansion of (1 + 2x)^n</a>
%H A317495 Zagros Lalo, <a href="/A317495/a317495_1.pdf">Second layer of skew diagonals in center-justified triangle of coefficients in expansion of (2 + x)^n</a>
%F A317495 T(n,k) = 2^(n - 3k) / ((n - 3k)! k!) * (n - 2k)! where n >= 0 and k = 0..floor(n/3).
%e A317495 Triangle begins:
%e A317495        1;
%e A317495        2;
%e A317495        4;
%e A317495        8,      1;
%e A317495       16,      4;
%e A317495       32,     12;
%e A317495       64,     32,      1;
%e A317495      128,     80,      6;
%e A317495      256,    192,     24;
%e A317495      512,    448,     80,      1;
%e A317495     1024,   1024,    240,      8;
%e A317495     2048,   2304,    672,     40;
%e A317495     4096,   5120,   1792,    160,     1;
%e A317495     8192,  11264,   4608,    560,    10;
%e A317495    16384,  24576,  11520,   1792,    60;
%e A317495    32768,  53248,  28160,   5376,   280,   1;
%e A317495    65536, 114688,  67584,  15360,  1120,  12;
%e A317495   131072, 245760, 159744,  42240,  4032,  84;
%e A317495   262144, 524288, 372736, 112640, 13440, 448, 1;
%t A317495 t[n_, k_] := t[n, k] = 2^(n - 3k)/((n - 3 k)! k!) (n - 2 k)!; Table[t[n, k], {n, 0, 18}, {k, 0, Floor[n/3]} ]  // Flatten
%t A317495 t[0, 0] = 1; t[n_, k_] := t[n, k] = If[n < 0 || k < 0, 0, 2 t[n - 1, k] + t[n - 3, k - 1]]; Table[t[n, k], {n, 0, 18}, {k, 0, Floor[n/3]}] // Flatten
%o A317495 (GAP) Flat(List([0..20],n->List([0..Int(n/3)],k->2^(n-3*k)/(Factorial(n-3*k)*Factorial(k))*Factorial(n-2*k)))); # _Muniru A Asiru_, Jul 31 2018
%o A317495 (Magma) /* As triangle */ [[2^(n-3*k)/(Factorial(n-3*k)*Factorial(k))* Factorial(n-2*k): k in [0..Floor(n/3)]]: n in [0.. 15]]; // _Vincenzo Librandi_, Sep 05 2018
%Y A317495 Row sums give A008998, A020708.
%Y A317495 Cf. A013609
%Y A317495 Cf. A038207
%Y A317495 Cf. A317494
%Y A317495 Cf. A128099, A207538.
%Y A317495 Cf. A000079 (column 0), A001787 (column 1), A001788 (column 2), A001789 (column 3), A003472 (column 4).
%K A317495 tabf,nonn,easy
%O A317495 0,2
%A A317495 _Zagros Lalo_, Jul 30 2018