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A213849 Rectangular array: (row n) = b**c, where b(h) = ceiling(h/2), c(h) = floor(n-1+h), n>=1, h>=1, and ** = convolution.

%I A213849 #10 Jul 13 2012 11:46:48
%S A213849 1,2,1,5,3,2,8,6,4,2,14,11,9,5,3,20,17,14,10,6,3,30,26,23,17,13,7,4,
%T A213849 40,36,32,26,20,14,8,4,55,50,46,38,32,23,17,9,5,70,65,60,52,44,35,26,
%U A213849 18,10,5,91,85,80,70,62,50,41,29,21,11,6
%N A213849 Rectangular array:  (row n) = b**c, where b(h) = ceiling(h/2), c(h) = floor(n-1+h), n>=1, h>=1, and ** = convolution.
%C A213849 Principal diagonal: A049778.
%C A213849 Antidiagonal sums: A213850.
%C A213849 Row 1, (1,1,2,2,3,3,...)**(1,1,2,2,3,3,...).
%C A213849 Row 2, (1,1,2,2,3,3,...)**(1,2,2,3,3,4,...).
%C A213849 Row 3, (1,1,2,2,3,3,...)**(2,2,3,3,4,4,...).
%C A213849 For a guide to related arrays, see A212500.
%H A213849 Clark Kimberling, <a href="/A213849/b213849.txt">Antidiagonals n = 1..60, flattened</a>
%F A213849 T(n,k) = 4*T(n,k-1)-6*T(n,k-2)+4*T(n,k-3)-T(n,k-4).
%F A213849 G.f. for row n: f(x)/g(x), where f(x) = x*(ceiling(n/2) + m(n)*x - floor(n/2)*x^2), where m(n) = (n+1 mod 2), and g(x) = (1+x)^2 *(1-x)^4.
%e A213849 Northwest corner (the array is read by falling antidiagonals):
%e A213849 1...2...5....8....14...20...30...40
%e A213849 1...3...6....11...17...26...36...50
%e A213849 2...4...9....14...23...32...46...60
%e A213849 2...5...10...17...26...38...52...70
%e A213849 3...6...13...20...32...44...62...80
%t A213849 b[n_]:=Floor[(n+1)/2];c[n_]:=Floor[(n+1)/2];
%t A213849 t[n_,k_]:=Sum[b[k-i]c[n+i],{i,0,k-1}]
%t A213849 TableForm[Table[t[n,k],{n,1,10},{k,1,10}]]
%t A213849 Flatten[Table[t[n-k+1,k],{n,12},{k,n,1,-1}]]
%t A213849 r[n_]:=Table[t[n,k],{k,1,60}]  (* A213849 *)
%t A213849 d=Table[t[n,n],{n,1,50}] (* A049778 *)
%t A213849 s[n_]:=Sum[t[i,n+1-i],{i,1,n}]
%t A213849 s1=Table[s[n],{n,1,50}] (* A213850 *)
%Y A213849 Cf. A212500.
%K A213849 nonn,tabl,easy
%O A213849 1,2
%A A213849 _Clark Kimberling_, Jul 05 2012