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A213841 Rectangular array: (row n) = b**c, where b(h) = 2*h-1, c(h) = 4*n-7+4*h, n>=1, h>=1, and ** = convolution.

%I A213841 #9 Jul 13 2012 11:41:43
%S A213841 1,8,5,29,24,9,72,65,40,13,145,136,101,56,17,256,245,200,137,72,21,
%T A213841 413,400,345,264,173,88,25,624,609,544,445,328,209,104,29,897,880,805,
%U A213841 688,545,392,245,120,33,1240,1221,1136,1001
%N A213841 Rectangular array:  (row n) = b**c, where b(h) = 2*h-1, c(h) = 4*n-7+4*h, n>=1, h>=1, and ** = convolution.
%C A213841 Principal diagonal: A213842.
%C A213841 Antidiagonal sums: A213843.
%C A213841 Row 1, (1,5,9,13,...)**(1,3,5,7,...): A100178.
%C A213841 Row 2, (1,5,9,13,...)**(3,5,7,9,...): (4*k^3 + 9*k^2 + 2*k)/3.
%C A213841 Row 3, (1,5,9,13,...)**(5,7,9,11,...): (4*k^3 + 21*k^2 + 2*k)/3.
%C A213841 For a guide to related arrays, see A212500.
%H A213841 Clark Kimberling, <a href="/A213841/b213841.txt">Antidiagonals n = 1..60, flattened</a>
%F A213841 T(n,k) = 4*T(n,k-1)-6*T(n,k-2)+4*T(n,k-3)-T(n,k-4).
%F A213841 G.f. for row n: f(x)/g(x), where f(x) = x*(4*n-3 + 4*x - (4*n-7)*x^2) and g(x) = (1-x)^4.
%e A213841 Northwest corner (the array is read by falling antidiagonals):
%e A213841 1....8....29....72....145
%e A213841 5....24...65....136...245
%e A213841 9....40...101...200...345
%e A213841 13...56...137...264...445
%e A213841 17...72...173...328...545
%e A213841 21...88...209...392...645
%t A213841 b[n_]:=2n-1;c[n_]:=4n-3;
%t A213841 t[n_,k_]:=Sum[b[k-i]c[n+i],{i,0,k-1}]
%t A213841 TableForm[Table[t[n,k],{n,1,10},{k,1,10}]]
%t A213841 Flatten[Table[t[n-k+1,k],{n,12},{k,n,1,-1}]]
%t A213841 r[n_]:=Table[t[n,k],{k,1,60}] (* A213841 *)
%t A213841 Table[t[n,n],{n,1,40}] (* A213842 *)
%t A213841 s[n_]:=Sum[t[i,n+1-i],{i,1,n}]
%t A213841 Table[s[n],{n,1,50}] (* A213843 *)
%Y A213841 Cf. A212500.
%K A213841 nonn,tabl,easy
%O A213841 1,2
%A A213841 _Clark Kimberling_, Jul 05 2012