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%I #6 Jul 12 2012 12:26:51
%S 1,6,2,18,11,3,40,30,16,4,75,62,42,21,5,126,110,84,54,26,6,196,177,
%T 145,106,66,31,7,288,266,228,180,128,78,36,8,405,380,336,279,215,150,
%U 90,41,9,550,522,472,406,330,250,172,102,46
%N Rectangular array: (row n) = b**c, where b(h) = 3*h-2, c(h) = n-1+h, n>=1, h>=1, and ** = convolution.
%C Principal diagonal: A213772
%C Antidiagonal sums: A132117
%C Row 1, (1,4,7,10,...)**(1,2,3,4,...): A002411
%C Row 2, (1,4,7,10,...)**(2,3,4,5,...): A162260
%C Row 3, (1,2,3,4,5,...)**(7,10,13,16,...): (k^3 + 7*k^2 - 2*k)/2
%C Row 4, (1,2,3,4,5,...)**(10,13,16,...): (k^3 + 10*k^2 - 3*k)/2
%C For a guide to related arrays, see A212500.
%H Clark Kimberling, <a href="/A213771/b213771.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*(n + (n+1)*x - (n+2)*x^2) and g(x) = (1 - x)^4.
%e Northwest corner (the array is read by falling antidiagonals):
%e 1....6....18...40....75....126
%e 2....11...30...62....110...177
%e 3....16...42...84....145...228
%e 4....21...54...106...180...279
%e 5....26...66...128...215...330
%t b[n_]:=3n-2;c[n_]:=n;
%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}] (* A213771 *)
%t Table[t[n,n],{n,1,40}] (* A213772 *)
%t s[n_]:=Sum[t[i,n+1-i],{i,1,n}]
%t Table[s[n],{n,1,50}] (* A132117 *)
%Y Cf. A212500
%K nonn,tabl,easy
%O 1,2
%A _Clark Kimberling_, Jul 04 2012
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