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A305834 Triangle read by rows: T(0,0)= 1; T(n,k)= T(n-1,k) + 4*T(n-2,k-1) for k = 0..floor(n/2); T(n,k)=0 for n or k < 0. 1
1, 1, 1, 4, 1, 8, 1, 12, 16, 1, 16, 48, 1, 20, 96, 64, 1, 24, 160, 256, 1, 28, 240, 640, 256, 1, 32, 336, 1280, 1280, 1, 36, 448, 2240, 3840, 1024, 1, 40, 576, 3584, 8960, 6144, 1, 44, 720, 5376, 17920, 21504, 4096 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,4
COMMENTS
The numbers in rows of the triangle are along skew diagonals pointing top-right in center-justified triangle given in A013611 ((1+4*x)^n).
The coefficients in the expansion of 1/(1-x-4*x^2) 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 2.5615528128...: A222132 (sqrt(4 + sqrt(4 + sqrt(4 + sqrt(4 + ... ))))), 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. 70, 72, 371, 372.
LINKS
FORMULA
G.f.: 1/(1 - t*x - 4*t^2).
Column k is binomial (n + k - 1, k) * 4^k.
EXAMPLE
Triangle begins:
1;
1;
1, 4;
1, 8;
1, 12, 16;
1, 16, 48;
1, 20, 96, 64;
1, 24, 160, 256;
1, 28, 240, 640, 256;
1, 32, 336, 1280, 1280;
1, 36, 448, 2240, 3840, 1024;
1, 40, 576, 3584, 8960, 6144;
1, 44, 720, 5376, 17920, 21504, 4096;
1, 48, 880, 7680, 32256, 57344, 28672;
1, 52, 1056, 10560, 53760, 129024, 114688, 16384;
1, 56, 1248, 14080, 84480, 258048, 344064, 131072;
1, 60, 1456, 18304, 126720, 473088, 860160, 589824, 65536;
1, 64, 1680, 23296, 183040, 811008, 1892352, 1966080, 589824;
MATHEMATICA
t[0, 0] = 1; t[n_, k_] := If[n < 0 || k < 0, 0, t[n - 1, k] + 4 t[n - 2, k - 1]]; Table[t[n, k], {n, 0, 12}, {k, 0, Floor[n/2]}] // Flatten
CROSSREFS
Row sums give A006131.
Cf. A000012 (column 0), A008586 (column 1), A035008 (column 2), A141478 (column 3), A120054 (column 4).
Cf. A013611.
Cf. A222132.
Sequence in context: A019425 A255242 A329371 * A295786 A080102 A106475
KEYWORD
tabf,nonn,easy
AUTHOR
Shara Lalo, Jun 11 2018
STATUS
approved

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Last modified April 20 07:43 EDT 2024. Contains 371799 sequences. (Running on oeis4.)