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A213768 Rectangular array: (row n) = b**c, where b(h) = F(h), c(h) = 2*n-3+2*h, F=A000045 (Fibonacci numbers), n>=1, h>=1, and ** = convolution.

%I A213768 #16 Jul 10 2019 10:47:09
%S A213768 1,4,3,10,8,5,21,18,12,7,40,35,26,16,9,72,64,49,34,20,11,125,112,88,
%T A213768 63,42,24,13,212,191,152,112,77,50,28,15,354,320,257,192,136,91,58,32,
%U A213768 17,585,530,428,323,232,160,105,66,36,19
%N A213768 Rectangular array:  (row n) = b**c, where b(h) = F(h), c(h) = 2*n-3+2*h, F=A000045 (Fibonacci numbers), n>=1, h>=1, and ** = convolution.
%C A213768 Principal diagonal:  A213769.
%C A213768 Antidiagonal sums:  A213770.
%C A213768 Row 1,  (1,1,2,3,5,...)**(1,3,5,7,9,...): A001891.
%C A213768 Row 2,  (1,1,2,3,5,...)**(3,5,7,9,11,...).
%C A213768 Row 3,  (1,1,2,3,5,...)**(5,7,9,11,13,...).
%C A213768 For a guide to related arrays, see A213500.
%H A213768 Clark Kimberling, <a href="/A213768/b213768.txt">Antidiagonals n=1..80, flattened</a>
%F A213768 T(n,k) = 3*T(n,k-1)-2*T(n,k-2)-T(n,k-3)+T(n,k-4).
%F A213768 G.f. for row n:  f(x)/g(x), where f(x) = x*(2*n - 1 - (2*n - 3)*x) and g(x) = (1 - x - x^2)(1 - x )^2.
%F A213768 T(n,k) = 2*n*Fibonacci(k+2) + Lucas(k+2) - 2*(k+n) - 3. - _Ehren Metcalfe_, Jul 08 2019
%e A213768 Northwest corner (the array is read by falling antidiagonals):
%e A213768 1....4....10...21...40....72....125
%e A213768 3....8....18...35...64....112...191
%e A213768 5....12...26...49...88....152...257
%e A213768 7....16...34...63...112...192...323
%e A213768 9....20...42...77...136...232...389
%e A213768 11...24...50...91...160...272...455
%t A213768 b[n_] := Fibonacci[n]; c[n_] := 2 n - 1;
%t A213768 t[n_, k_] := Sum[b[k - i] c[n + i], {i, 0, k - 1}]
%t A213768 TableForm[Table[t[n, k], {n, 1, 10}, {k, 1, 10}]]
%t A213768 Flatten[Table[t[n - k + 1, k], {n, 12}, {k, n, 1, -1}]]
%t A213768 r[n_] := Table[t[n, k], {k, 1, 60}]  (* A213768 *)
%t A213768 Table[t[n, n], {n, 1, 40}] (* A213769 *)
%t A213768 s[n_] := Sum[t[i, n + 1 - i], {i, 1, n}]
%t A213768 Table[s[n], {n, 1, 50}] (* A213770 *)
%Y A213768 Cf. A001891, A213500, A213769, A213770.
%K A213768 nonn,tabl,easy
%O A213768 1,2
%A A213768 _Clark Kimberling_, Jun 21 2012