A305837 Triangle read by rows: T(0,0) = 1; T(n,k) = 5*T(n-1,k) + T(n-2,k-1) for k = 0..floor(n/2); T(n,k)=0 for n or k < 0.

1, 5, 25, 1, 125, 10, 625, 75, 1, 3125, 500, 15, 15625, 3125, 150, 1, 78125, 18750, 1250, 20, 390625, 109375, 9375, 250, 1, 1953125, 625000, 65625, 2500, 25, 9765625, 3515625, 437500, 21875, 375, 1, 48828125, 19531250, 2812500, 175000, 4375, 30, 244140625, 107421875, 17578125, 1312500, 43750, 525, 1

Offset

0,2

Comments

The numbers in rows of the triangle are along skew diagonals pointing top-left in center-justified triangle given in A013612 ((1+5*x)^n).

The coefficients in the expansion of 1/(1-5x-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 5.1925824035..., a metallic mean (see A098318), 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, 92, 380, 382.

Links

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

Shara Lalo, Left-justified triangle

Shara Lalo, Skew diagonals in triangle A013612

Formula

G.f.: 1/(1 - 5*t*x - t^2).

Example

Triangle begins:
            1;
            5;
           25,           1;
          125,          10;
          625,          75,          1;
         3125,         500,         15;
        15625,        3125,        150,         1;
        78125,       18750,       1250,        20;
       390625,      109375,       9375,       250,        1;
      1953125,      625000,      65625,      2500,       25;
      9765625,     3515625,     437500,     21875,      375,      1;
     48828125,    19531250,    2812500,    175000,     4375,     30;
    244140625,   107421875,   17578125,   1312500,    43750,    525,     1;
   1220703125,   585937500,  107421875,   9375000,   393750,   7000,    35;
   6103515625,  3173828125,  644531250,  64453125,  3281250,  78750,   700,  1;
  30517578125, 17089843750, 3808593750, 429687500, 25781250, 787500, 10500, 40;

Mathematica

t[0, 0] = 1; t[n_, k_] := If[n < 0 || k < 0, 0, 5 t[n - 1, k] + t[n - 2, k - 1]]; Table[t[n, k], {n, 0, 12}, {k, 0, Floor[n/2]}] // Flatten

Crossrefs

Row sums give A052918.

Cf. A000351 (column 0), A053464 (column 1), A081135 (column 2), A081143 (column 3), A036071 (column 4).

Cf. A013612.

Cf. A098318.

Sequence in context: A142725 A270590 A271378 * A175555 A346994 A070387

Adjacent sequences: A305834 A305835 A305836 * A305838 A305839 A305840

Keyword

tabf,nonn,easy

Author

Shara Lalo, Jun 11 2018

Status

approved