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A213783 Rectangular array: (row n) = b**c, where b(h) = 1+[h/2], c(h) = [(n+h)/2], n>=1, h>=1, [ ] = floor, and ** = convolution.

%I A213783 #10 Jul 12 2012 12:13:57
%S A213783 1,3,1,6,4,2,11,8,6,2,17,14,11,7,3,26,22,19,13,9,3,36,32,28,22,16,10,
%T A213783 4,50,45,41,33,27,18,12,4,65,60,55,47,39,30,21,13,5,85,79,74,64,56,44,
%U A213783 35,23,15,5,106,100,94,84,74,62,50,38,26,16,6,133,126,120,108
%N A213783 Rectangular array:  (row n) = b**c, where b(h) = 1+[h/2], c(h) = [(n+h)/2], n>=1, h>=1, [ ] = floor, and ** = convolution.
%C A213783 Principal diagonal: A213759.
%C A213783 Antidiagonal sums: A213760.
%C A213783 Row 1, (1,2,2,3,3,4,4,...)**(1,1,2,2,3,3,4,...): A005744.
%C A213783 Row 2, (1,2,2,3,3,4,4,5,...)**(1,2,2,3,3,4,4,5,...).
%C A213783 Row 3, (1,2,2,3,3,4,4,5,...)**(2,2,3,3,4,4,5,5,...).
%C A213783 For a guide to related arrays, see A213500.
%H A213783 Clark Kimberling, <a href="/A213783/b213783.txt">Antidiagonals n = 1..80, flattened</a>
%F A213783 T(n,k) = 2*T(n,k-1) -T(n,k-2) - 4*T(n,k-3) +T(n,k-4) +2*T(n,k-5) -T(n,k-6).
%F A213783 G.f. for row n:  f(x)/g(x), where f(x) = [(n+1)/2] + [(n+2)/2]*x + ([(n-1)/2] + [(n+1)/2])*x^2 - (1+[n/2]-(n mod 2))*x^3 + [n/2]*x^4 and g(x) = (1 + x)^2 *(1 - x)^4, where [ ] = floor.
%e A213783 Northwest corner (the array is read by falling antidiagonals):
%e A213783 1...3....6....11...17...26...36....50
%e A213783 1...4....8....14...22...32...45....60
%e A213783 2...6....11...19...28...41...55....74
%e A213783 2...7....13...22...33...47...64....84
%e A213783 3...9....16...27...39...56...74....98
%e A213783 3...10...18...30...44...62...83....108
%e A213783 4...12...21...35...50...71...93....122
%e A213783 4...13...23...38...55...77...102...132
%t A213783 b[n_] := Floor[(n + 2)/2]; c[n_] := Floor[(n + 1)/2];
%t A213783 t[n_, k_] := Sum[b[k - i] c[n + i], {i, 0, k - 1}]
%t A213783 TableForm[Table[t[n, k], {n, 1, 10}, {k, 1, 10}]]
%t A213783 Flatten[Table[t[n - k + 1, k], {n, 12}, {k, n, 1, -1}]]
%t A213783 r[n_] := Table[t[n, k], {k, 1, 60}]  (* A213783 *)
%t A213783 Table[t[n, n], {n, 1, 40}] (* A213759 *)
%t A213783 s[n_] := Sum[t[i, n + 1 - i], {i, 1, n}]
%t A213783 Table[s[n], {n, 1, 50}] (* A213760 *)
%Y A213783 Cf. A213500.
%K A213783 nonn,tabl,easy
%O A213783 1,2
%A A213783 _Clark Kimberling_, Jun 22 2012