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 A318775 Triangle read by rows: T(0,0) = 1; T(n,k) = T(n-1,k) + 2 * T(n-5,k-1) for k = 0..floor(n/5); T(n,k)=0 for n or k < 0. 2
 1, 1, 1, 1, 1, 1, 2, 1, 4, 1, 6, 1, 8, 1, 10, 1, 12, 4, 1, 14, 12, 1, 16, 24, 1, 18, 40, 1, 20, 60, 1, 22, 84, 8, 1, 24, 112, 32, 1, 26, 144, 80, 1, 28, 180, 160, 1, 30, 220, 280, 1, 32, 264, 448, 16, 1, 34, 312, 672, 80, 1, 36, 364, 960, 240, 1, 38, 420, 1320, 560, 1, 40, 480, 1760, 1120 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,7 COMMENTS The numbers in rows of the triangle are along a "fourth layer" skew diagonals pointing top-right in center-justified triangle given in A013609 ((1+2*x)^n) and along a "fourth layer" 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+2x)^n and (2+x)^n are given in A128099 and A207538 respectively.) The coefficients in the expansion of 1/(1-x-2x^5) are given by the sequence generated by the row sums. The row sums give A318777. If s(n) is the row sum at n, then the ratio s(n)/s(n-1) is approximately 1.4510850920547191..., 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. LINKS FORMULA T(n,k) = 2^k / ((n - 5*k)! k!) * (n - 4*k)! where n >= 0 and 0 <= k <= floor(n/5). EXAMPLE Triangle begins:   1;   1;   1;   1;   1;   1,  2;   1,  4;   1,  6;   1,  8;   1, 10;   1, 12,   4;   1, 14,  12;   1, 16,  24;   1, 18,  40;   1, 20,  60;   1, 22,  84,    8;   1, 24, 112,   32;   1, 26, 144,   80;   1, 28, 180,  160;   1, 30, 220,  280;   1, 32, 264,  448,  16;   1, 34, 312,  672,  80;   1, 36, 364,  960, 240;   1, 38, 420, 1320, 560;   ... MATHEMATICA t[n_, k_] := t[n, k] = 2^k/((n - 5 k)! k!) (n - 4 k)!; Table[t[n, k], {n, 0, 24}, {k, 0, Floor[n/5]} ]  // Flatten. t[0, 0] = 1; t[n_, k_] := t[n, k] = If[n < 0 || k < 0, 0, t[n - 1, k] + 2 t[n - 5, k - 1]]; Table[t[n, k], {n, 0, 24}, {k, 0, Floor[n/5]}] // Flatten. CROSSREFS Row sums give A318777. Cf. A013609, A038207, A128099, A207538. Sequence in context: A214060 A009531 A124625 * A317500 A317494 A317505 Adjacent sequences:  A318772 A318773 A318774 * A318776 A318777 A318778 KEYWORD tabf,nonn,easy AUTHOR Zagros Lalo, Sep 04 2018 STATUS approved

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Last modified May 19 07:05 EDT 2019. Contains 323386 sequences. (Running on oeis4.)