|
%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
|
