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, 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

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

Table of n, a(n) for n=0..48.

Shara Lalo, Right justified triangle

Shara Lalo, Skew diagonals in triangle A013611

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

Adjacent sequences: A305831 A305832 A305833 * A305835 A305836 A305837

Keyword

tabf,nonn,easy

Author

Shara Lalo, Jun 11 2018

Status

approved