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A318772 Triangle read by rows: T(0,0) = 1; T(n,k) = T(n-1,k) + 3 * T(n-4,k-1) for k = 0..floor(n/4); T(n,k)=0 for n or k < 0. 2
1, 1, 1, 1, 1, 3, 1, 6, 1, 9, 1, 12, 1, 15, 9, 1, 18, 27, 1, 21, 54, 1, 24, 90, 1, 27, 135, 27, 1, 30, 189, 108, 1, 33, 252, 270, 1, 36, 324, 540, 1, 39, 405, 945, 81, 1, 42, 495, 1512, 405, 1, 45, 594, 2268, 1215, 1, 48, 702, 3240, 2835, 1, 51, 819, 4455, 5670, 243, 1, 54, 945, 5940, 10206, 1458 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,6

COMMENTS

The numbers in rows of the triangle are along a "third layer" skew diagonals pointing top-right in center-justified triangle given in A013610 ((1+3*x)^n) and along a "third layer" 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^4) are given by the sequence generated by the row sums.

The row sums give A318774.

If s(n) is the row sum at n, then the ratio s(n)/s(n-1) is approximately 1.6580980673722..., 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.

LINKS

Table of n, a(n) for n=0..71.

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

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

FORMULA

T(n,k) = 3^k / ((n - 4*k)! k!) * (n - 3*k)! where n >= 0 and 0 <= k <= floor(n/4).

EXAMPLE

Triangle begins:

  1;

  1;

  1;

  1;

  1,  3;

  1,  6;

  1,  9;

  1, 12;

  1, 15,   9;

  1, 18,  27;

  1, 21,  54;

  1, 24,  90;

  1, 27, 135,   27;

  1, 30, 189,  108;

  1, 33, 252,  270;

  1, 36, 324,  540;

  1, 39, 405,  945,   81;

  1, 42, 495, 1512,  405;

  1, 45, 594, 2268, 1215;

  ...

MATHEMATICA

t[n_, k_] := t[n, k] = 3^k/((n - 4 k)! k!) (n - 3 k)!; Table[t[n, k], {n, 0, 21}, {k, 0, Floor[n/4]} ]  // Flatten.

t[0, 0] = 1; t[n_, k_] := t[n, k] = If[n < 0 || k < 0, 0, t[n - 1, k] + 3 t[n - 4, k - 1]]; Table[t[n, k], {n, 0, 21}, {k, 0, Floor[n/4]}] // Flatten.

CROSSREFS

Row sums give A318774.

Cf. A013610, A027465.

Cf. A304236, A304249.

Cf. A317496, A317497.

Sequence in context: A168111 A109646 A199783 * A317496 A304236 A145063

Adjacent sequences:  A318769 A318770 A318771 * A318773 A318774 A318775

KEYWORD

tabf,nonn,easy

AUTHOR

Zagros Lalo, Sep 04 2018

STATUS

approved

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Last modified June 25 22:25 EDT 2019. Contains 324361 sequences. (Running on oeis4.)