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A213761 Rectangular array: (row n) = b**c, where b(h) = h, c(h) = 3*n-5+3*h, n>=1, h>=1, and ** = convolution.

%I A213761 #12 Feb 14 2014 04:13:25
%S A213761 1,6,4,18,15,7,40,36,24,10,75,70,54,33,13,126,120,100,72,42,16,196,
%T A213761 189,165,130,90,51,19,288,280,252,210,160,108,60,22,405,396,364,315,
%U A213761 255,190,126,69,25,550,540,504,448,378,300
%N A213761 Rectangular array:  (row n) = b**c, where b(h) = h, c(h) = 3*n-5+3*h, n>=1, h>=1, and ** = convolution.
%C A213761 Principal diagonal: A172073.
%C A213761 Antidiagonal sums: A002419.
%C A213761 Row 1, (1,2,3,4,5,...)**(1,4,7,10,13,...): A002411.
%C A213761 Row 2, (1,2,3,4,5,...)**(4,7,10,13,16,...): A077414.
%C A213761 Row 3, (1,2,3,4,5,...)**(7,10,13,16,...): (k^3 + 7*k^2 + 6*k)/2.
%C A213761 Row 4, (1,2,3,4,5,...)**(10,13,16,...): (k^3 + 10*k^2 + 9*k)/2.
%C A213761 For a guide to related arrays, see A212500.
%H A213761 Clark Kimberling, <a href="/A213761/b213761.txt">Antidiagonals n = 1..45, flattened</a>
%F A213761 T(n,k) = 4*T(n,k-1)-6*T(n,k-2)+4*T(n,k-3)-T(n,k-4).
%F A213761 G.f. for row n: f(x)/g(x), where f(x) = x*(3*n - 2 - (3*n - 5)*x) and g(x) = (1 - x)^4.
%e A213761 Northwest corner (the array is read by falling antidiagonals):
%e A213761 1....6....18...40....75....126
%e A213761 4....15...36...70....120...189
%e A213761 7....24...54...100...165...252
%e A213761 10...33...72...130...210...315
%e A213761 13...42...90...160...255...378
%t A213761 b[n_]:=n;c[n_]:=3n-2;
%t A213761 t[n_,k_]:=Sum[b[k-i]c[n+i],{i,0,k-1}]
%t A213761 TableForm[Table[t[n,k],{n,1,10},{k,1,10}]]
%t A213761 Flatten[Table[t[n-k+1,k],{n,12},{k,n,1,-1}]]
%t A213761 r[n_]:=Table[t[n,k],{k,1,60}] (* A213761 *)
%t A213761 Table[t[n,n],{n,1,40}] (* A172073 *)
%t A213761 s[n_]:=Sum[t[i,n+1-i],{i,1,n}]
%t A213761 Table[s[n],{n,1,50}] (* A002419 *)
%Y A213761 Cf. A212500.
%K A213761 nonn,tabl,easy
%O A213761 1,2
%A A213761 _Clark Kimberling_, Jul 04 2012