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 A013610 ((1+3*x)^n) and along a "second layer" of 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^3) 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 1.863706527819..., 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. 364-366.
LINKS
FORMULA
T(n,k) = 3^k * (n-2*k)!/ (k! * (n-3*k)!) where n is a nonnegative integer and k = 0..floor(n/3).
EXAMPLE
Triangle begins:
1;
1;
1;
1, 3;
1, 6;
1, 9;
1, 12, 9;
1, 15, 27;
1, 18, 54;
1, 21, 90, 27;
1, 24, 135, 108;
1, 27, 189, 270;
1, 30, 252, 540, 81;
1, 33, 324, 945, 405;
1, 36, 405, 1512, 1215;
1, 39, 495, 2268, 2835, 243;
1, 42, 594, 3240, 5670, 1458;
1, 45, 702, 4455, 10206, 5103;
1, 48, 819, 5940, 17010, 13608, 729;
MATHEMATICA
T[n_, k_]:= T[n, k] = 3^k*(n-2*k)!/((n-3*k)!*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] + 3T[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->3^k/(Factorial(n-3*k)*Factorial(k))*Factorial(n-2*k)))); # Muniru A Asiru, Aug 01 2018
(Magma) [3^k*Binomial(n-2*k, k): k in [0..Floor(n/3)], n in [0..24]]; // G. C. Greubel, May 12 2021
(Sage) flatten([[3^k*binomial(n-2*k, k) for k in (0..n//3)] for n in (0..24)]) # G. C. Greubel, May 12 2021
CROSSREFS
KEYWORD
tabf,nonn,easy
AUTHOR
Zagros Lalo, Jul 31 2018
STATUS
approved