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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. 2
 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, 80, 1, 20, 112, 160, 16, 1, 22, 144, 280, 80, 1, 24, 180, 448, 240, 1, 26, 220, 672, 560, 32, 1, 28, 264, 960, 1120, 192, 1, 30, 312, 1320, 2016, 672, 1, 32, 364, 1760, 3360, 1792, 64 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,5 COMMENTS 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.) The coefficients in the expansion of 1/(1-x-2x^3) are given by the sequence generated by the row sums. The row sums give A003229. 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. REFERENCES Shara Lalo and Zagros Lalo, Polynomial Expansion Theorems and Number Triangles, Zana Publishing, 2018, ISBN: 978-1-9995914-0-3, pp. 358, 359 LINKS FORMULA T(n,k) = 2^k / ((n - 3k)! k!) * (n - 2k)! where n is a nonnegative integer and k = 0..floor(n/3). EXAMPLE Triangle begins:   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,   80;   1, 20, 112,  160,   16;   1, 22, 144,  280,   80;   1, 24, 180,  448,  240;   1, 26, 220,  672,  560,   32;   1, 28, 264,  960, 1120,  192;   1, 30, 312, 1320, 2016,  672;   1, 32, 364, 1760, 3360, 1792, 64; MATHEMATICA 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[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. PROG (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 CROSSREFS Row sums give A003229. Cf. A013609, A038207, A289265, A317495, A128099, A207538. Sequence in context: A124625 A318775 A317500 * A317505 A137374 A131516 Adjacent sequences:  A317491 A317492 A317493 * A317495 A317496 A317497 KEYWORD tabf,nonn,easy AUTHOR Zagros Lalo, Jul 30 2018 STATUS approved

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Last modified May 19 10:36 EDT 2019. Contains 323390 sequences. (Running on oeis4.)