OFFSET
0,6
COMMENTS
The numbers in rows of the triangle are along a "third layer" skew diagonals pointing top-right in center-justified triangle given in A013610 ((1+3*x)^n) and along a "third layer" skew diagonals pointing top-left in center-justified triangle given in A027465 ((3+x)^n), see links. (Note: First layer of skew diagonals in center-justified triangles of coefficients in expansions of (1+3*x)^n and (3+x)^n are given in A304236 and A304249 respectively.)
The coefficients in the expansion of 1/(1-x-3x^4) are given by the sequence generated by the row sums.
If s(n) is the row sum at n, then the ratio s(n)/s(n-1) is approximately 1.6580980673722..., 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) = 3^k * (n - 3*k)!/ ((n - 4*k)! k!), where n >= 0 and 0 <= k <= floor(n/4).
EXAMPLE
Triangle begins:
1;
1;
1;
1;
1, 3;
1, 6;
1, 9;
1, 12;
1, 15, 9;
1, 18, 27;
1, 21, 54;
1, 24, 90;
1, 27, 135, 27;
1, 30, 189, 108;
1, 33, 252, 270;
1, 36, 324, 540;
1, 39, 405, 945, 81;
1, 42, 495, 1512, 405;
1, 45, 594, 2268, 1215;
...
MATHEMATICA
T[n_, k_]:= T[n, k]= 3^k(n-3k)!/((n-4k)! k!); Table[T[n, k], {n, 0, 21}, {k, 0, Floor[n/4]} ] // Flatten.
T[0, 0] = 1; T[n_, k_] := T[n, k] = If[n<0 || k<0, 0, T[n-1, k] + 3T[n-4, k-1]]; Table[T[n, k], {n, 0, 21}, {k, 0, Floor[n/4]}] // Flatten.
PROG
(Magma) [3^k*Binomial(n-3*k, k): k in [0..Floor(n/4)], n in [0..24]]; // G. C. Greubel, May 12 2021
(Sage) flatten([[3^k*binomial(n-3*k, k) for k in (0..n//4)] for n in (0..24)]) # G. C. Greubel, May 12 2021
CROSSREFS
KEYWORD
tabf,nonn,easy
AUTHOR
Zagros Lalo, Sep 04 2018
STATUS
approved