A305695 - OEIS

%I #49 Aug 19 2018 06:28:44

%S 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,

%T 14,1,54,441,1431,2126,1205,287,22,1,88,1155,6072,14502,13581,4928,

%U 736,35,1,143,3025,25707,99587,149717,90013,20371,1905,56,1

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

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

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

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

%e n\k|   0    1     2     3      4     5     6    7  8 9

%e ---+--------------------------------------------------

%e 0  |   1

%e 1  |   2    1

%e 2  |   4    3     1

%e 3  |   7    9     4     1

%e 4  |  12   24    19     6      1

%e 5  |  20   64    79    46      9     1

%e 6  |  33  168   339   306    113    14     1

%e 7  |  54  441  1431  2126   1205   287    22    1

%e 8  |  88 1155  6072 14502  13581  4928   736   35  1

%e 9  | 143 3025 25707 99587 149717 90013 20371 1905 56 1

%o (PARI) f(n, k) = prod(j=0, k-1, fibonacci(n-j))/prod(j=1, k, fibonacci(j));

%o T(n, k) = if (n< 0, 0, T(n-1, k) + f(n+1, k+1));

%o tabl(nn) = for (n=0, nn, for (k=0, n, print1(T(n, k), ", ")); print); \\ _Michel Marcus_, Jul 20 2018

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

%K nonn,tabl

%O 0,2

%A _Tony Foster III_, Jul 09 2018

%E More terms from _Michel Marcus_, Jul 20 2018