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A318772
Triangle read by rows: T(0,0) = 1; T(n,k) = T(n-1,k) + 3 * T(n-4,k-1) for k = 0..floor(n/4); T(n,k)=0 for n or k < 0.
4
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, 1, 48, 702, 3240, 2835, 1, 51, 819, 4455, 5670, 243, 1, 54, 945, 5940, 10206, 1458
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.
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
Row sums give A318774.
Sequence in context: A109646 A199783 A329645 * A317496 A304236 A360654
KEYWORD
tabf,nonn,easy
AUTHOR
Zagros Lalo, Sep 04 2018
STATUS
approved