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.
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.
LINKS
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
KEYWORD
tabf,nonn,easy
AUTHOR
Zagros Lalo, May 08 2018
STATUS
approved