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%I #9 Jul 13 2012 11:43:33
%S 3,16,9,47,36,15,104,89,56,21,195,176,131,76,27,328,305,248,173,96,33,
%T 511,484,415,320,215,116,39,752,721,640,525,392,257,136,45,1059,1024,
%U 931,796,635,464,299,156,51,1440,1401
%N Rectangular array: (row n) = b**c, where b(h) = 4*h-1, c(h) = 2*n-3+2*h, n>=1, h>=1, and ** = convolution.
%C Principal diagonal: A213848.
%C Antidiagonal sums: A180324.
%C Row 1, (3,7,11,15,...)**(1,3,5,7,...): A172482.
%C Row 2, (3,7,11,15,...)**(3,5,7,9,...): (4*k^3 + 15*k^2 + 8*k)/3.
%C Row 3, (3,7,11,15,...)**(5,7,9,13,...): (4*k^3 + 27*k^2 + 14*k)/3.
%C For a guide to related arrays, see A212500.
%H Clark Kimberling, <a href="/A213847/b213847.txt">Antidiagonals n = 1..60, flattened</a>
%F T(n,k) = 4*T(n,k-1)-6*T(n,k-2)+4*T(n,k-3)-T(n,k-4).
%F G.f. for row n: f(x)/g(x), where f(x) = x*(6*n-3 + 4*(n-2)x - (2*n-3)*x^2) and g(x) = (1-x)^4.
%e Northwest corner (the array is read by falling antidiagonals):
%e 3....16...47....104...195...328
%e 9....36...89....176...305...484
%e 15...56...131...248...415...640
%e 21...76...173...320...525...796
%t b[n_]:=4n-1;c[n_]:=2n-1;
%t t[n_,k_]:=Sum[b[k-i]c[n+i],{i,0,k-1}]
%t TableForm[Table[t[n,k],{n,1,10},{k,1,10}]]
%t Flatten[Table[t[n-k+1,k],{n,12},{k,n,1,-1}]]
%t r[n_]:=Table[t[n,k],{k,1,60}] (* A213847 *)
%t Table[t[n,n],{n,1,40}] (* A213848 *)
%t s[n_]:=Sum[t[i,n+1-i],{i,1,n}]
%t Table[s[n],{n,1,50}] (* A180324 *)
%Y Cf. A212500.
%K nonn,tabl,easy
%O 1,1
%A _Clark Kimberling_, Jul 05 2012
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