A305695 Triangle T(n,k) read by rows: fibonomial coefficients sums triangle.

1, 2, 1, 4, 3, 1, 7, 9, 4, 1, 12, 24, 19, 6, 1, 20, 64, 79, 46, 9, 1, 33, 168, 339, 306, 113, 14, 1, 54, 441, 1431, 2126, 1205, 287, 22, 1, 88, 1155, 6072, 14502, 13581, 4928, 736, 35, 1, 143, 3025, 25707, 99587, 149717, 90013, 20371, 1905, 56, 1

Offset

0,2

Comments

The triangle coefficients give sums of Fibonacci powers when multiplied with Lang triangle coefficients and summed (see 2nd formula).

Links

Table of n, a(n) for n=0..54.

Formula

T(n, k) = T(n-1, k) + A010048(n+1, k+1).

Sum_{t=0..n-1} A056588(n-1, n-1-t) * T(k+t, n-1) = Sum_{j=1..k+1} F(j)^n.

Example

n\k|   0    1     2     3      4     5     6    7  8 9
---+--------------------------------------------------
0  |   1
1  |   2    1
2  |   4    3     1
3  |   7    9     4     1
4  |  12   24    19     6      1
5  |  20   64    79    46      9     1
6  |  33  168   339   306    113    14     1
7  |  54  441  1431  2126   1205   287    22    1
8  |  88 1155  6072 14502  13581  4928   736   35  1
9  | 143 3025 25707 99587 149717 90013 20371 1905 56 1

Prog

(PARI) f(n, k) = prod(j=0, k-1, fibonacci(n-j))/prod(j=1, k, fibonacci(j));
T(n, k) = if (n< 0, 0, T(n-1, k) + f(n+1, k+1));
tabl(nn) = for (n=0, nn, for (k=0, n, print1(T(n, k), ", ")); print); \\ Michel Marcus, Jul 20 2018

Crossrefs

Cf. A000045, A010048, A000071, A056588, A317360.

Sequence in context: A131254 A210229 A210213 * A211235 A134626 A347633

Adjacent sequences: A305692 A305693 A305694 * A305696 A305697 A305698

Keyword

nonn,tabl

Author

Tony Foster III, Jul 09 2018

Extensions

More terms from Michel Marcus, Jul 20 2018

Status

approved