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A213777 Rectangular array: (row n) = b**c, where b(h) = F(h), c(h) = F(h+1), F=A000045 (Fibonacci numbers), n>=1, h>=1, and ** = convolution.

%I A213777 #13 Jul 10 2019 08:22:26
%S A213777 1,3,2,7,5,3,15,12,8,5,30,25,19,13,8,58,50,40,31,21,13,109,96,80,65,
%T A213777 50,34,21,201,180,154,130,105,81,55,34,365,331,289,250,210,170,131,89,
%U A213777 55,655,600,532,469,404,340,275,212,144,89,1164,1075,965,863
%N A213777 Rectangular array:  (row n) = b**c, where b(h) = F(h), c(h) = F(h+1), F=A000045 (Fibonacci numbers), n>=1, h>=1, and ** = convolution.
%C A213777 Principal diagonal:  A001870
%C A213777 Antidiagonal sums:  A152881
%C A213777 row 1,  (1,1,2,3,5,8,...)**(1,2,3,5,8,13,...): A023610(k-1)
%C A213777 row 2,  (1,1,2,3,5,8,...)**(2,3,5,8,13,21,...): A067331(k-1)
%C A213777 row 3,  (1,1,2,3,5,8,...)**(3,5,8,13,21,34,...)
%C A213777 For a guide to related arrays, see A213500.
%H A213777 Clark Kimberling, <a href="/A213777/b213777.txt">Antidiagonals n=1..80, flattened</a>
%F A213777 T(n,k) = 2*T(n,k-1) + T(n,k-2) - 2*T(n,k-3) - T(n,k-4).
%F A213777 G.f. for row n:  f(x)/g(x), where f(x) = F(n-1) + F(n-2)*x and g(x) = (1 - x - x^2)^2.
%F A213777 T(n,k) = (k*Lucas(n+k+1) + Lucas(n)*Fibonacci(k))/5. - _Ehren Metcalfe_, Jul 10 2019
%e A213777 Northwest corner (the array is read by falling antidiagonals):
%e A213777 1....3....7....15....30....58
%e A213777 2....5....12...25....50....96
%e A213777 3....8....19...40....80....154
%e A213777 5....13...31...65....130...250
%e A213777 8....21...50...105...210...404
%e A213777 13...34...81...170...340...654
%t A213777 b[n_] := Fibonacci[n]; c[n_] := Fibonacci[n + 1];
%t A213777 t[n_, k_] := Sum[b[k - i] c[n + i], {i, 0, k - 1}]
%t A213777 TableForm[Table[t[n, k], {n, 1, 10}, {k, 1, 10}]]
%t A213777 Flatten[Table[t[n - k + 1, k], {n, 12}, {k, n, 1, -1}]]
%t A213777 r[n_] := Table[t[n, k], {k, 1, 60}]  (* A213777 *)
%t A213777 Table[t[n, n], {n, 1, 40}] (* A001870 *)
%t A213777 s[n_] := Sum[t[i, n + 1 - i], {i, 1, n}]
%t A213777 Table[s[n], {n, 1, 50}] (* A152881 *)
%Y A213777 Cf. A213500.
%K A213777 nonn,tabl,easy
%O A213777 1,2
%A A213777 _Clark Kimberling_, Jun 21 2012