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A213838 Rectangular array: (row n) = b**c, where b(h) = 4*h-3, c(h) = 2*n-3+2*h, n>=1, h>=1, and ** = convolution.

%I A213838 #11 Jul 11 2012 04:30:23
%S A213838 1,8,3,29,20,5,72,59,32,7,145,128,89,44,9,256,235,184,119,56,11,413,
%T A213838 388,325,240,149,68,13,624,595,520,415,296,179,80,15,897,864,777,652,
%U A213838 505,352,209,92,17,1240,1203,1104,959,784
%N A213838 Rectangular array:  (row n) = b**c, where b(h) = 4*h-3, c(h) = 2*n-3+2*h, n>=1, h>=1, and ** = convolution.
%C A213838 Principal diagonal: A213839.
%C A213838 Antidiagonal sums: A213840.
%C A213838 Row 1, (1,5,9,13,...)**(1,3,5,7,...): A100178.
%C A213838 Row 2, (1,5,9,13,...)**(3,5,7,9,...): (4*k^3 + 9*k^2 - 4*k)/3.
%C A213838 Row 3, (1,5,9,13,...)**(5,7,9,11,...): (4*k^3 + 21*k^2 - 10*k)/3.
%C A213838 For a guide to related arrays, see A212500.
%H A213838 Clark Kimberling, <a href="/A213838/b213838.txt">Antidiagonals n = 1..60, flattened</a>
%F A213838 T(n,k) = 4*T(n,k-1)-6*T(n,k-2)+4*T(n,k-3)-T(n,k-4).
%F A213838 G.f. for row n: f(x)/g(x), where f(x) = x*(2*n-1 + 4*n*x - (6*n-9)*x^2) and g(x) = (1-x)^4.
%e A213838 Northwest corner (the array is read by falling antidiagonals):
%e A213838 1....8....29....72....145
%e A213838 3....20...59....128...235
%e A213838 5....32...89....184...325
%e A213838 7....44...119...240...415
%e A213838 9....56...149...296...505
%e A213838 11...68...179...352...595
%t A213838 b[n_]:=4n-3; c[n_]:=2n-1;
%t A213838 t[n_,k_]:=Sum[b[k-i]c[n+i],{i,0,k-1}]
%t A213838 TableForm[Table[t[n,k],{n,1,10},{k,1,10}]]
%t A213838 Flatten[Table[t[n-k+1,k],{n,12},{k,n,1,-1}]]
%t A213838 r[n_]:=Table[t[n,k],{k,1,60}] (* A213838 *)
%t A213838 Table[t[n,n],{n,1,40}] (* A213839 *)
%t A213838 s[n_]:=Sum[t[i,n+1-i],{i,1,n}]
%t A213838 Table[s[n],{n,1,50}] (* A213840 *)
%Y A213838 Cf. A212500.
%K A213838 nonn,tabl,easy
%O A213838 1,2
%A A213838 _Clark Kimberling_, Jul 05 2012