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A317496 Triangle read by rows: T(0,0) = 1; T(n,k) = T(n-1,k) + 3 * T(n-3,k-1) for k = 0..floor(n/3); T(n,k)=0 for n or k < 0. 4
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 (list; graph; refs; listen; history; text; internal format)
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.

The row sums give A084386.

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

Table of n, a(n) for n=0..69.

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 - 3k)! k!) * (n - 2k)! 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 - 3 k)! k!) (n - 2 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] + 3 t[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

CROSSREFS

Row sums give A084386.

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

Sequence in context: A199783 A329645 A318772 * A304236 A145063 A202851

Adjacent sequences:  A317493 A317494 A317495 * A317497 A317498 A317499

KEYWORD

tabf,nonn,easy

AUTHOR

Zagros Lalo, Jul 31 2018

STATUS

approved

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Last modified February 28 20:48 EST 2021. Contains 341731 sequences. (Running on oeis4.)