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A318773
Triangle T(n,k) = 3*T(n-1,k) + T(n-4,k-1) for k = 0..floor(n/4), with T(0,0) = 1 and T(n,k) = 0 for n or k < 0, read by rows.
4
1, 3, 9, 27, 81, 1, 243, 6, 729, 27, 2187, 108, 6561, 405, 1, 19683, 1458, 9, 59049, 5103, 54, 177147, 17496, 270, 531441, 59049, 1215, 1, 1594323, 196830, 5103, 12, 4782969, 649539, 20412, 90, 14348907, 2125764, 78732, 540, 43046721, 6908733, 295245, 2835, 1, 129140163, 22320522, 1082565, 13608, 15
OFFSET
0,2
COMMENTS
The numbers in rows of the triangle are along a "third layer" skew diagonals pointing top-left in center-justified triangle given in A013610 ((1+3*x)^n) and along a "third layer" skew diagonals pointing top-right 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/(1-3x-x^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 3.035744112294..., 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^(n-4*k) * (n-3*k)!/(k! * (n-4*k)!) where n >= 0 and 0 <= k <= floor(n/4).
EXAMPLE
Triangle begins:
1;
3;
9;
27;
81, 1;
243, 6;
729, 27;
2187, 108;
6561, 405, 1;
19683, 1458, 9;
59049, 5103, 54;
177147, 17496, 270;
531441, 59049, 1215, 1;
1594323, 196830, 5103, 12;
4782969, 649539, 20412, 90;
14348907, 2125764, 78732, 540;
43046721, 6908733, 295245, 2835, 1;
129140163, 22320522, 1082565, 13608, 15;
387420489, 71744535, 3897234, 61236, 135;
...
MATHEMATICA
T[n_, k_]:= T[n, k] = 3^(n-4k)*(n-3k)!/((n-4k)! k!); Table[T[n, k], {n, 0, 16}, {k, 0, Floor[n/4]} ]//Flatten.
T[0, 0] = 1; T[n_, k_]:= T[n, k] = If[n<0 || k<0, 0, 3T[n-1, k] + T[n-4, k-1]]; Table[T[n, k], {n, 0, 16}, {k, 0, Floor[n/4]}]//Flatten.
PROG
(Magma) [3^(n-4*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^(n-4*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 A052917.
Cf. A000244 (column 0), A027471 (column 1), A027472 (column 2), A036216 (column 3).
Sequences of the form 3^(n-q*k)*binomial(n-(q-1)*k, k): A027465 (q=1), A304249 (q=2), A317497 (q=3), this sequence (q=4).
Sequence in context: A291006 A289781 A209239 * A331631 A036142 A036160
KEYWORD
tabf,nonn,easy
AUTHOR
Zagros Lalo, Sep 04 2018
STATUS
approved