A304236 Triangle T(n,k) = T(n-1,k) + 3*T(n-2,k-1) for k = 0..floor(n/2), with T(0,0) = 1, T(n,k) = 0 for n or k < 0, read by rows.

1, 1, 1, 3, 1, 6, 1, 9, 9, 1, 12, 27, 1, 15, 54, 27, 1, 18, 90, 108, 1, 21, 135, 270, 81, 1, 24, 189, 540, 405, 1, 27, 252, 945, 1215, 243, 1, 30, 324, 1512, 2835, 1458, 1, 33, 405, 2268, 5670, 5103, 729, 1, 36, 495, 3240, 10206, 13608, 5103, 1, 39, 594, 4455, 17010, 30618, 20412, 2187

Offset

0,4

Comments

The numbers in rows of the triangle are along skew diagonals pointing top-right in center-justified triangle given in A013610 ((1+3*x)^n).

The coefficients in the expansion of 1/(1-x-3x^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, 88, 363.

Links

G. C. Greubel, Rows n = 0..100 of the irregular triangle, flattened

Zagros Lalo, Left-justified triangle

Zagros Lalo, Skew diagonals in center-justified triangle of coefficients in expansion of (1+3x)^n

Formula

T(n,k) = 3^k*binomial(n-k,k), n >= 0, 0 <= k <= floor(n/2).

Example

Triangle begins:
  1;
  1;
  1,  3;
  1,  6;
  1,  9,   9;
  1, 12,  27;
  1, 15,  54,   27;
  1, 18,  90,  108;
  1, 21, 135,  270,    81;
  1, 24, 189,  540,   405;
  1, 27, 252,  945,  1215,    243;
  1, 30, 324, 1512,  2835,   1458;
  1, 33, 405, 2268,  5670,   5103,    729;
  1, 36, 495, 3240, 10206,  13608,   5103;
  1, 39, 594, 4455, 17010,  30618,  20412,   2187;
  1, 42, 702, 5940, 26730,  61236,  61236,  17496;
  1, 45, 819, 7722, 40095, 112266, 153090,  78732,  6561;
  1, 48, 945, 9828, 57915, 192456, 336798, 262440, 59049;

Maple

seq(seq( 3^k*binomial(n-k, k), k=0..floor(n/2)), n=0..24); # G. C. Greubel, May 12 2021

Mathematica

T[0, 0] = 1; T[n_, k_]:= If[n<0 || k<0, 0, T[n-1, k] + 3*T[n-2, k-1]]; Table[T[n, k], {n, 0, 12}, {k, 0, Floor[n/2]}]//Flatten.
Table[3^k Binomial[n-k, k], {n, 0, 17}, {k, 0, Floor[n/2]}]//Flatten.

Prog

(PARI) T(n, k) = if ((n<0) || (k<0), 0, if ((n==0) && (k==0), 1, T(n-1, k) + 3*T(n-2, k-1)));
tabf(nn) = for (n=0, nn, for (k=0, n\2, print1(T(n, k), ", ")); print); \\ Michel Marcus, May 10 2018
(Magma) /* As triangle */ [[3^k*Binomial(n-k, k): k in [0..Floor(n/2)]]: n in [0.. 15]]; // Vincenzo Librandi, Sep 05 2018
(Sage) flatten([[3^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 A006130.

Cf. A013610.

Sequences of the form 3^k*binomial(n-(q-1)*k, k): A013610 (q=1), this sequence (q=2), A317496 (q=3), A318772 (q=4).

Sequence in context: A329645 A318772 A317496 * A360654 A145063 A202851

Adjacent sequences: A304233 A304234 A304235 * A304237 A304238 A304239

Keyword

tabf,nonn,easy

Author

Zagros Lalo, May 08 2018

Status

approved