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

Author

Zagros Lalo, May 08 2018

Keywords

• tabf
• nonn
• easy

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.

Examples

			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;
		

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

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).

Programs

Magma

/* As triangle */ [[3^k*Binomial(n-k,k): k in [0..Floor(n/2)]]: n in [0.. 15]]; // Vincenzo Librandi, Sep 05 2018

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, 14}, {k, 0, Floor[n/2]}]//Flatten
Table[3^k Binomial[n-k, k], {n, 0, 14}, {k, 0, Floor[n/2]}]//Flatten

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

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

Formula

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