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A318773 Triangle read by rows: T(0,0) = 1; T(n,k) = 3 T(n-1,k) + T(n-4,k-1) for k = 0..floor(n/4); T(n,k)=0 for n or k < 0. 2
1, 3, 9, 27, 81, 1, 243, 6, 729, 27, 2187, 108, 6561, 405, 1, 19683, 1458, 9, 59049, 5103, 54, 177147, 17496, 270, 531441, 59049, 1215, 1, 1594323, 196830, 5103, 12, 4782969, 649539, 20412, 90, 14348907, 2125764, 78732, 540, 43046721, 6908733, 295245, 2835, 1, 129140163, 22320522, 1082565, 13608, 15 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

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

The row sums give A052917.

If s(n) is the row sum at n, then the ratio s(n)/s(n-1) is approximately 3.035744112294..., 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..49.

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

EXAMPLE

Triangle begins:

          1;

          3;

          9;

         27;

         81,        1;

        243,        6;

        729,       27;

       2187,      108;

       6561,      405,       1;

      19683,     1458,       9;

      59049,     5103,      54;

     177147,    17496,     270;

     531441,    59049,    1215,     1;

    1594323,   196830,    5103,    12;

    4782969,   649539,   20412,    90;

   14348907,  2125764,   78732,   540;

   43046721,  6908733,  295245,  2835,   1;

  129140163, 22320522, 1082565, 13608,  15;

  387420489, 71744535, 3897234, 61236, 135;

  ...

MATHEMATICA

t[n_, k_] := t[n, k] = 3^(n - 4 k)/((n - 4 k)! k!) (n - 3 k)!; Table[t[n, k], {n, 0, 16}, {k, 0, Floor[n/4]} ]  // Flatten.

t[0, 0] = 1; t[n_, k_] := t[n, k] = If[n < 0 || k < 0, 0, 3 t[n - 1, k] + t[n - 4, k - 1]]; Table[t[n, k], {n, 0, 16}, {k, 0, Floor[n/4]}] // Flatten.

CROSSREFS

Row sums give A052917.

Cf. A013610, A027465.

Cf. A304236, A304249

Cf. A317496, A317497.

Cf. A000244 (column 0), A027471 (column 1), A027472 (column 2), A036216 (column 3).

Sequence in context: A291006 A289781 A209239 * A036142 A036160 A271352

Adjacent sequences:  A318770 A318771 A318772 * A318774 A318775 A318776

KEYWORD

tabf,nonn,easy

AUTHOR

Zagros Lalo, Sep 04 2018

STATUS

approved

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Last modified June 19 16:48 EDT 2019. Contains 324222 sequences. (Running on oeis4.)