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A213819 Rectangular array: (row n) = b**c, where b(h) = h, c(h) = 3*n-4+3*h, n>=1, h>=1, and ** = convolution.

%I A213819 #13 Jul 16 2012 05:06:42
%S A213819 2,9,5,24,18,8,50,42,27,11,90,80,60,36,14,147,135,110,78,45,17,224,
%T A213819 210,180,140,96,54,20,324,308,273,225,170,114,63,23,450,432,392,336,
%U A213819 270,200,132,72,26,605,585,540,476,399,315
%N A213819 Rectangular array:  (row n) = b**c, where b(h) = h, c(h) = 3*n-4+3*h, n>=1, h>=1, and ** = convolution.
%C A213819 Principal diagonal: A213820.
%C A213819 Antidiagonal sums: A153978.
%C A213819 Row 1, (1,2,3,4,...)**(2,5,8,11,...): A006002.
%C A213819 Row 2, (1,2,3,4,...)**(5,8,11,14,...): is it the sequence A212343?.
%C A213819 Row 3, (1,2,3,4,...)**(8,11,14,17,...): (k^3 + 8*k^2 + 7*k)/2.
%C A213819 For a guide to related arrays, see A212500.
%H A213819 Clark Kimberling, <a href="/A213819/b213819.txt">Antidiagonals n = 1..60, flattened</a>
%F A213819 T(n,k) = 4*T(n,k-1)-6*T(n,k-2)+4*T(n,k-3)-T(n,k-4).
%F A213819 G.f. for row n: f(x)/g(x), where f(x) = x(3*n-1 - (3*n-4)*x) and g(x) = (1-x)^4.
%e A213819 Northwest corner (the array is read by falling antidiagonals):
%e A213819 2....9....24....50....90....147
%e A213819 5....18...42....80....135...210
%e A213819 8....27...60....110...180...273
%e A213819 11...36...78....140...225...336
%e A213819 14...45...96....170...270...399
%e A213819 17...54...114...200...315...462
%t A213819 b[n_]:=n;c[n_]:=3n-1;
%t A213819 t[n_,k_]:=Sum[b[k-i]c[n+i],{i,0,k-1}]
%t A213819 TableForm[Table[t[n,k],{n,1,10},{k,1,10}]]
%t A213819 Flatten[Table[t[n-k+1,k],{n,12},{k,n,1,-1}]]
%t A213819 r[n_]:=Table[t[n,k],{k,1,60}] (* A213819 *)
%t A213819 Table[t[n,n],{n,1,40}] (* A213820 *)
%t A213819 d/2 (* A002414 *)
%t A213819 s[n_]:=Sum[t[i,n+1-i],{i,1,n}]
%t A213819 Table[s[n],{n,1,50}] (* A153978 *)
%t A213819 s1/2 (* A001296 *)
%Y A213819 Cf. A212500
%K A213819 nonn,tabl,easy
%O A213819 1,1
%A A213819 _Clark Kimberling_, Jul 04 2012