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A214985 Array: T(m,n) = (F(n) + F(2*n) + ... + F(n*m))/F(n), by antidiagonals; transpose of A214984.

%I A214985 #12 Jan 09 2024 16:40:01
%S A214985 1,1,2,1,4,4,1,5,12,7,1,8,22,33,12,1,12,56,94,88,20,1,19,134,385,399,
%T A214985 232,33,1,30,342,1487,2640,1691,609,54,1,48,872,6138,16492,18096,7164,
%U A214985 1596,88,1,77,2256,25319,110143,182900,124033,30348,4180,143
%N A214985 Array: T(m,n) = (F(n) + F(2*n) + ... + F(n*m))/F(n), by antidiagonals; transpose of A214984.
%C A214985 row 1: A001612 (except for initial term)
%C A214985 col 1: A000071
%C A214985 col 2: A027941
%C A214985 col 3: A049652
%C A214985 col 4: A092521
%C A214985 col 6: A049664
%C A214985 col 8: A156093 without minus signs
%H A214985 Clark Kimberling, <a href="/A214985/b214985.txt">Antidiagonals n = 1..60, flattened</a>
%F A214985 For odd-numbered columns (m odd):
%F A214985 T(m,n) = (F(m*n+m) + F(m*n) - F(m))/(F(m)*L(m)).
%F A214985 For even-numbered columns (m even):
%F A214985 T(m,n) = (F(m*n+m) - F(m*n) - F(m))/(F(m)*(L(m)-1)).
%e A214985 Northwest corner:
%e A214985 1....1.....1......1.......1
%e A214985 2....4.....5......8.......12
%e A214985 4....12....22.....56......134
%e A214985 7....33....94.....385.....1487
%e A214985 12...88....399....2640....16492
%e A214985 20...232...1691...18096...182900
%t A214985 F[n_] := Fibonacci[n]; L[n_] := LucasL[n];
%t A214985 t[m_, n_] := (1/F[n])*Sum[F[k*n], {k, 1, m}]
%t A214985 TableForm[Table[t[m, n], {m, 1, 10}, {n, 1, 10}]]
%t A214985 Flatten[Table[t[k, n + 1 - k], {n, 1, 12}, {k, 1, n}]]
%Y A214985 Cf. A214978, A214984, A214986.
%K A214985 nonn,tabl
%O A214985 1,3
%A A214985 _Clark Kimberling_, Oct 28 2012