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

Author

Tony Foster III, Jul 09 2018

Keywords

• nonn
• tabl

Comments

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

Examples

			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
		

Crossrefs

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

Programs

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

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.

Extensions

More terms from Michel Marcus, Jul 20 2018