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A305695
Triangle T(n,k) read by rows: fibonomial coefficients sums triangle.
1
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).
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
KEYWORD
nonn,tabl
AUTHOR
Tony Foster III, Jul 09 2018
EXTENSIONS
More terms from Michel Marcus, Jul 20 2018
STATUS
approved