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A213765 Rectangular array: (row n) = b**c, where b(h) = 2*n-1, c(h) = F(n-1+h), F=A000045 (Fibonacci numbers), n>=1, h>=1, and ** = convolution.

%I A213765 #18 Jul 10 2019 11:16:53
%S A213765 1,4,1,10,5,2,21,14,9,3,40,31,24,14,5,72,61,52,38,23,8,125,112,101,83,
%T A213765 62,37,13,212,197,184,162,135,100,60,21,354,337,322,296,263,218,162,
%U A213765 97,34,585,566,549,519,480,425,353,262,157,55,960,939,920,886
%N A213765 Rectangular array:  (row n) = b**c, where b(h) = 2*n-1, c(h) = F(n-1+h), F=A000045 (Fibonacci numbers), n>=1, h>=1, and ** = convolution.
%C A213765 Principal diagonal: A213766.
%C A213765 Antidiagonal sums: A213767.
%C A213765 Row 1,  (1,3,5,7,9,...)**(1,1,2,3,5,...): A001891.
%C A213765 Row 2,  (1,3,5,7,9,...)**(1,2,3,5,8,...): A023652.
%C A213765 Row 3,  (1,3,5,7,9,...)**(2,3,5,8,13,...).
%C A213765 For a guide to related arrays, see A213500.
%H A213765 Clark Kimberling, <a href="/A213765/b213765.txt">Antidiagonals n = 1..60, flattened</a>
%F A213765 T(n,k) = 3*T(n,k-1)-2*T(n,k-2)-T(n,k-3)+T(n,k-4).
%F A213765 G.f. for row n:  f(x)/g(x), where f(x) = x*(F(n) + F(n+1)*x - F(n-1)*x^2) and g(x) = (1 - x - x^2)(1 - x )^2.
%F A213765 T(n,k) = F(n+k+4) - 2*k*F(n+1) - F(n+4), F = A000045. - _Ehren Metcalfe_, Jul 10 2019
%e A213765 Northwest corner (the array is read by falling antidiagonals):
%e A213765 1....4....10....21....40....72
%e A213765 1....5....14....31....61....112
%e A213765 2....9....24....52....101...184
%e A213765 3....14...38....83....162...296
%e A213765 5....23...62....135...263...480
%e A213765 8....37...100...218...425...776
%e A213765 13...60...162...353...688...1256
%t A213765 b[n_] := 2 n - 1; c[n_] := Fibonacci[n];
%t A213765 t[n_, k_] := Sum[b[k - i] c[n + i], {i, 0, k - 1}]
%t A213765 TableForm[Table[t[n, k], {n, 1, 10}, {k, 1, 10}]]
%t A213765 Flatten[Table[t[n - k + 1, k], {n, 12}, {k, n, 1, -1}]]
%t A213765 r[n_] := Table[t[n, k], {k, 1, 60}]  (* A213765 *)
%t A213765 Table[t[n, n], {n, 1, 40}] (* A213766 *)
%t A213765 s[n_] := Sum[t[i, n + 1 - i], {i, 1, n}]
%t A213765 Table[s[n], {n, 1, 50}] (* A213767 *)
%Y A213765 Cf. A213500.
%K A213765 nonn,tabl,easy
%O A213765 1,2
%A A213765 _Clark Kimberling_, Jun 21 2012