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A304249
Triangle T(n,k) = 3*T(n-1,k) + T(n-2,k-1) for k = 0..floor(n/2), with T(0,0) = 1 and T(n,k) = 0 for n < 0 or k < 0, read by rows.
6
1, 3, 9, 1, 27, 6, 81, 27, 1, 243, 108, 9, 729, 405, 54, 1, 2187, 1458, 270, 12, 6561, 5103, 1215, 90, 1, 19683, 17496, 5103, 540, 15, 59049, 59049, 20412, 2835, 135, 1, 177147, 196830, 78732, 13608, 945, 18, 531441, 649539, 295245, 61236, 5670, 189, 1
OFFSET
0,2
COMMENTS
The numbers in rows of the triangle are along skew diagonals pointing top-left in center-justified triangle given in A013610 ((1+3*x)^n), cf. formula.
The coefficients in the expansion of 1/(1-3x-x^2) are given by the sequence generated by the row sums.
The sequence of the row sums are the "Bronze Fibonacci numbers" A006190, and the limit of their ratio is 3.30277563773... (Bronze ratio), see A098316.
REFERENCES
Shara Lalo and Zagros Lalo, Polynomial Expansion Theorems and Number Triangles, Zana Publishing, 2018, ISBN: 978-1-9995914-0-3, pp. 70, 72, 86, 363.
FORMULA
T(n,k) = A013610(n-k, n-2k). - M. F. Hasler, Jun 01 2018
EXAMPLE
Triangle begins:
1;
3;
9, 1;
27, 6;
81, 27, 1;
243, 108, 9;
729, 405, 54, 1;
2187, 1458, 270, 12;
6561, 5103, 1215, 90, 1;
19683, 17496, 5103, 540, 15;
59049, 59049, 20412, 2835, 135, 1;
177147, 196830, 78732, 13608, 945, 18;
531441, 649539, 295245, 61236, 5670, 189, 1;
1594323, 2125764, 1082565, 262440, 30618, 1512, 21;
4782969, 6908733, 3897234, 1082565, 153090, 10206, 252, 1;
14348907, 22320522, 13817466, 4330260, 721710, 61236, 2268, 24;
43046721, 71744535, 48361131, 16888014, 3247695, 336798, 17010, 324, 1;
129140163, 229582512, 167403915, 64481508, 14073345, 1732104, 112266, 3240, 27;
MATHEMATICA
T[0, 0] = 1; T[n_, k_]:= If[n<0 || k<0, 0, 3T[n-1, k] + T[n-2, k-1]];
Table[t[n, k], {n, 0, 12}, {k, 0, Floor[n/2]}]//Flatten
With[{q=2}, Table[3^(n-q*k)*Binomial[n-(q-1)*k, k], {n, 0, 24}, {k, 0, Floor[n/q]}] ]//Flatten (* G. C. Greubel, May 12 2021 *)
PROG
(PARI) T(n, k)=if( n>0 && k>0, 3*T(n-1, k) + T(n-2, k-1), !n && !k)
tabf(nn) = for (n=0, nn, for (k=0, n\2, print1(T(n, k), ", ")); print); \\ Michel Marcus, May 10 2018
(Magma) [3^(n-2*k)*Binomial(n-k, k): k in [0..Floor(n/2)], n in [0..24]]; // G. C. Greubel, May 12 2021
(Sage) flatten([[3^(n-2*k)*binomial(n-k, k) for k in (0..n//2)] for n in (0..24)]) # G. C. Greubel, May 12 2021
CROSSREFS
Row sums give A006190.
Sequences of the form 3^(n-q*k)*binomial(n-(q-1)*k, k): A027465 (q=1), this sequence (q=2), A317497 (q=3), A318773 (q=4).
Sequence in context: A329214 A080322 A126179 * A128727 A126177 A158483
KEYWORD
tabf,nonn,easy
AUTHOR
Zagros Lalo, May 08 2018
STATUS
approved