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A213844 Rectangular array: (row n) = b**c, where b(h) = 2*h-1, c(h) = 4*n-5+4*h, n>=1, h>=1, and ** = convolution.

%I A213844 #9 Jul 13 2012 11:42:50
%S A213844 3,16,7,47,32,11,104,83,48,15,195,168,119,64,19,328,295,232,155,80,23,
%T A213844 511,472,395,296,191,96,27,752,707,616,495,360,227,112,31,1059,1008,
%U A213844 903,760,595,424,263,128,35,1440,1383,1264
%N A213844 Rectangular array:  (row n) = b**c, where b(h) = 2*h-1, c(h) = 4*n-5+4*h, n>=1, h>=1, and ** = convolution.
%C A213844 Principal diagonal: A213845.
%C A213844 Antidiagonal sums: A213846.
%C A213844 Row 1, (1,3,5,7...)**(3,7,11,15,...): A172482.
%C A213844 Row 2, (1,3,5,7,...)**(7,11,15,19,...): (4*k^3 + 15*k^2 + 2*k)/3.
%C A213844 Row 3, (1,3,5,7,...)**(11,15,19,23,...): (4*k^3 + 27*k^2 + 2*k)/3.
%C A213844 For a guide to related arrays, see A212500.
%H A213844 Clark Kimberling, <a href="/A213844/b213844.txt">Antidiagonals n = 1..60, flattened</a>
%F A213844 T(n,k) = 4*T(n,k-1)-6*T(n,k-2)+4*T(n,k-3)-T(n,k-4).
%F A213844 G.f. for row n: f(x)/g(x), where f(x) = x*(4*n-1 + 4*x - (4*n-5)*x^2) and g(x) = (1-x)^4.
%e A213844 Northwest corner (the array is read by falling antidiagonals):
%e A213844 3....16...47....104...195...328
%e A213844 7....32...83....168...295...472
%e A213844 11...48...119...232...395...616
%e A213844 15...64...155...296...495...760
%t A213844 b[n_]:=2n-1;c[n_]:=4n-1;
%t A213844 t[n_,k_]:=Sum[b[k-i]c[n+i],{i,0,k-1}]
%t A213844 TableForm[Table[t[n,k],{n,1,10},{k,1,10}]]
%t A213844 Flatten[Table[t[n-k+1,k],{n,12},{k,n,1,-1}]]
%t A213844 r[n_]:=Table[t[n,k],{k,1,60}] (* A213844 *)
%t A213844 Table[t[n,n],{n,1,40}] (* A213845 *)
%t A213844 s[n_]:=Sum[t[i,n+1-i],{i,1,n}]
%t A213844 Table[s[n],{n,1,50}] (* A213846 *)
%Y A213844 Cf. A212500.
%K A213844 nonn,tabl,easy
%O A213844 1,1
%A A213844 _Clark Kimberling_, Jul 05 2012