A317496 Triangle T(n,k) = T(n-1,k) + 3*T(n-3,k-1) for k = 0..floor(n/3) with T(0,0) = 1, T(n,k) = 0 for n or k < 0, read by rows.

1, 1, 1, 1, 3, 1, 6, 1, 9, 1, 12, 9, 1, 15, 27, 1, 18, 54, 1, 21, 90, 27, 1, 24, 135, 108, 1, 27, 189, 270, 1, 30, 252, 540, 81, 1, 33, 324, 945, 405, 1, 36, 405, 1512, 1215, 1, 39, 495, 2268, 2835, 243, 1, 42, 594, 3240, 5670, 1458, 1, 45, 702, 4455, 10206, 5103, 1, 48, 819, 5940, 17010, 13608, 729

Offset

0,5

Comments

The numbers in rows of the triangle are along a "second layer" of skew diagonals pointing top-right in center-justified triangle given in A013610 ((1+3*x)^n) and along a "second layer" of skew diagonals pointing top-left 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-x-3x^3) 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 1.863706527819..., 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, pp. 364-366.

Links

G. C. Greubel, Rows n = 0..120 of the irregular triangle, flattened

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

Zagros Lalo, Second layer skew diagonals in center-justified triangle of coefficients in expansion of (3 + x)^n

Formula

T(n,k) = 3^k * (n-2*k)!/ (k! * (n-3*k)!) where n is a nonnegative integer and k = 0..floor(n/3).

Example

Triangle begins:
  1;
  1;
  1;
  1,  3;
  1,  6;
  1,  9;
  1, 12,   9;
  1, 15,  27;
  1, 18,  54;
  1, 21,  90,   27;
  1, 24, 135,  108;
  1, 27, 189,  270;
  1, 30, 252,  540,    81;
  1, 33, 324,  945,   405;
  1, 36, 405, 1512,  1215;
  1, 39, 495, 2268,  2835,   243;
  1, 42, 594, 3240,  5670,  1458;
  1, 45, 702, 4455, 10206,  5103;
  1, 48, 819, 5940, 17010, 13608, 729;

Mathematica

T[n_, k_]:= T[n, k] = 3^k*(n-2*k)!/((n-3*k)!*k!); Table[T[n, k], {n, 0, 18}, {k, 0, Floor[n/3]} ]//Flatten
T[0, 0] = 1; T[n_, k_]:= T[n, k] = If[n<0 || k<0, 0, T[n-1, k] + 3T[n-3, k-1]]; Table[T[n, k], {n, 0, 18}, {k, 0, Floor[n/3]}]//Flatten

Prog

(GAP) Flat(List([0..20], n->List([0..Int(n/3)], k->3^k/(Factorial(n-3*k)*Factorial(k))*Factorial(n-2*k)))); # Muniru A Asiru, Aug 01 2018
(Magma) [3^k*Binomial(n-2*k, k): k in [0..Floor(n/3)], n in [0..24]]; // G. C. Greubel, May 12 2021
(Sage) flatten([[3^k*binomial(n-2*k, k) for k in (0..n//3)] for n in (0..24)]) # G. C. Greubel, May 12 2021

Crossrefs

Row sums give A084386.

Cf. A013610, A027465, A304236, A304249, A317497.

Sequences of the form 3^k*binomial(n-(q-1)*k, k): A013610 (q=1), A304236 (q=2), this sequence (q=3), A318772 (q=4).

Sequence in context: A199783 A329645 A318772 * A304236 A360654 A145063

Adjacent sequences: A317493 A317494 A317495 * A317497 A317498 A317499

Keyword

tabf,nonn,easy

Author

Zagros Lalo, Jul 31 2018

Status

approved