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

%I A305695 #49 Aug 19 2018 06:28:44
%S A305695 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 A305695 14,1,54,441,1431,2126,1205,287,22,1,88,1155,6072,14502,13581,4928,
%U A305695 736,35,1,143,3025,25707,99587,149717,90013,20371,1905,56,1
%N A305695 Triangle T(n,k) read by rows: fibonomial coefficients sums triangle.
%C A305695 The triangle coefficients give sums of Fibonacci powers when multiplied with Lang triangle coefficients and summed (see 2nd formula).
%F A305695 T(n, k) = T(n-1, k) + A010048(n+1, k+1).
%F A305695 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 A305695 n\k|   0    1     2     3      4     5     6    7  8 9
%e A305695 ---+--------------------------------------------------
%e A305695 0  |   1
%e A305695 1  |   2    1
%e A305695 2  |   4    3     1
%e A305695 3  |   7    9     4     1
%e A305695 4  |  12   24    19     6      1
%e A305695 5  |  20   64    79    46      9     1
%e A305695 6  |  33  168   339   306    113    14     1
%e A305695 7  |  54  441  1431  2126   1205   287    22    1
%e A305695 8  |  88 1155  6072 14502  13581  4928   736   35  1
%e A305695 9  | 143 3025 25707 99587 149717 90013 20371 1905 56 1
%o A305695 (PARI) f(n, k) = prod(j=0, k-1, fibonacci(n-j))/prod(j=1, k, fibonacci(j));
%o A305695 T(n, k) = if (n< 0, 0, T(n-1, k) + f(n+1, k+1));
%o A305695 tabl(nn) = for (n=0, nn, for (k=0, n, print1(T(n, k), ", ")); print); \\ _Michel Marcus_, Jul 20 2018
%Y A305695 Cf. A000045, A010048, A000071, A056588, A317360.
%K A305695 nonn,tabl
%O A305695 0,2
%A A305695 _Tony Foster III_, Jul 09 2018
%E A305695 More terms from _Michel Marcus_, Jul 20 2018