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A317494 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.

%I A317494 #20 Aug 08 2025 07:05:18
%S A317494 1,1,1,1,2,1,4,1,6,1,8,4,1,10,12,1,12,24,1,14,40,8,1,16,60,32,1,18,84,
%T A317494 80,1,20,112,160,16,1,22,144,280,80,1,24,180,448,240,1,26,220,672,560,
%U A317494 32,1,28,264,960,1120,192,1,30,312,1320,2016,672,1,32,364,1760,3360,1792,64
%N A317494 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 A317494 The numbers in rows of the triangle are along a "second layer" of skew diagonals pointing top-right in center-justified triangle given in A013609 ((1+2*x)^n) and along a "second layer" of skew diagonals pointing top-left 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 A317494 The coefficients in the expansion of 1/(1-x-2*x^3) are given by the sequence generated by the row sums.
%C A317494 The row sums give A003229.
%C A317494 If s(n) is the row sum at n, then the ratio s(n)/s(n-1) is approximately 1.695620769559862... (see A289265), when n approaches infinity.
%D A317494 Shara Lalo and Zagros Lalo, Polynomial Expansion Theorems and Number Triangles, Zana Publishing, 2018, ISBN: 978-1-9995914-0-3, pp. 358, 359
%H A317494 Zagros Lalo, <a href="/A317494/a317494.pdf">Second layer skew diagonals in center-justified triangle of coefficients in expansion of (2 + x)^n</a>
%H A317494 Zagros Lalo, <a href="/A317494/a317494_1.pdf">Second layer skew diagonals in center-justified triangle of coefficients in expansion of (1 + 2x)^n</a>
%F A317494 T(n,k) = 2^k / ((n - 3k)! k!) * (n - 2k)! where n is a nonnegative integer and k = 0..floor(n/3).
%e A317494 Triangle begins:
%e A317494   1;
%e A317494   1;
%e A317494   1;
%e A317494   1,  2;
%e A317494   1,  4;
%e A317494   1,  6;
%e A317494   1,  8,   4;
%e A317494   1, 10,  12;
%e A317494   1, 12,  24;
%e A317494   1, 14,  40,    8;
%e A317494   1, 16,  60,   32;
%e A317494   1, 18,  84,   80;
%e A317494   1, 20, 112,  160,   16;
%e A317494   1, 22, 144,  280,   80;
%e A317494   1, 24, 180,  448,  240;
%e A317494   1, 26, 220,  672,  560,   32;
%e A317494   1, 28, 264,  960, 1120,  192;
%e A317494   1, 30, 312, 1320, 2016,  672;
%e A317494   1, 32, 364, 1760, 3360, 1792, 64;
%t A317494 t[n_, k_] := t[n, k] = 2^k/((n - 3 k)! k!) (n - 2 k)!; Table[t[n, k], {n, 0, 18}, {k, 0, Floor[n/3]} ] // Flatten
%t A317494 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, 18}, {k, 0, Floor[n/3]}] // Flatten
%o A317494 (GAP) Flat(List([0..20],n->List([0..Int(n/3)],k->2^k/(Factorial(n-3*k)*Factorial(k))*Factorial(n-2*k)))); # _Muniru A Asiru_, Jul 31 2018
%Y A317494 Row sums give A003229.
%Y A317494 Cf. A013609, A038207, A289265, A317495, A128099, A207538.
%K A317494 tabf,nonn,easy
%O A317494 0,5
%A A317494 _Zagros Lalo_, Jul 30 2018