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