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A213825 Rectangular array: (row n) = b**c, where b(h) = 3*h-1, c(h) = 3*n-5+3*h, n>=1, h>=1, and ** = convolution.

%I A213825 #9 Jul 13 2012 09:49:05
%S A213825 2,13,8,42,34,14,98,87,55,20,190,176,132,76,26,327,310,254,177,97,32,
%T A213825 518,498,430,332,222,118,38,772,749,669,550,410,267,139,44,1098,1072,
%U A213825 980,840,670,488,312,160,50,1505,1476,1372
%N A213825 Rectangular array:  (row n) = b**c, where b(h) = 3*h-1, c(h) = 3*n-5+3*h, n>=1, h>=1, and ** = convolution.
%C A213825 Principal diagonal: A213826
%C A213825 Antidiagonal sums: A213827
%C A213825 Row 1, (2,5,8,13,...)**(1,4,7,10,13,...): (3*k^2 + k)/2
%C A213825 Row 2, (2,5,8,13,...)**(4,7,10,13,...): (3*k^3 + 9*k^2 - 2*k)/2
%C A213825 Row 3, (2,5,8,13,...)**(7,10,13,16,...): (3*k^3 + 18*k^2 - 5*k)/2
%C A213825 For a guide to related arrays, see A212500.
%H A213825 Clark Kimberling, <a href="/A213825/b213825.txt">Antidiagonals n = 1..80, flattened</a>
%F A213825 T(n,k) = 4*T(n,k-1)-6*T(n,k-2)+4*T(n,k-3)-T(n,k-4).
%F A213825 G.f. for row n: f(x)/g(x), where f(x) = x*((3*n-1) + (3*n+2)*x - (6*n-8)*x^2) and g(x) = (1-x)^4.
%e A213825 Northwest corner (the array is read by falling antidiagonals):
%e A213825 2....13....42....98....190
%e A213825 8....34....87....176...310
%e A213825 14...55....132...254...430
%e A213825 20...76....177...332...550
%e A213825 26...97....222...410...670
%e A213825 32...118...267...488...790
%t A213825 b[n_]:=3n-1;c[n_]:=3n-2;
%t A213825 t[n_,k_]:=Sum[b[k-i]c[n+i],{i,0,k-1}]
%t A213825 TableForm[Table[t[n,k],{n,1,10},{k,1,10}]]
%t A213825 Flatten[Table[t[n-k+1,k],{n,12},{k,n,1,-1}]]
%t A213825 r[n_]:=Table[t[n,k],{k,1,60}] (* A213825 *)
%t A213825 d=Table[t[n,n],{n,1,40}] (* A213826 *)
%t A213825 d/2 (* A024215 *)
%t A213825 s[n_]:=Sum[t[i,n+1-i],{i,1,n}]
%t A213825 s1=Table[s[n],{n,1,50}] (* A213827 *)
%Y A213825 Cf. A212500
%K A213825 nonn,tabl,easy
%O A213825 1,1
%A A213825 _Clark Kimberling_, Jul 04 2012