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A213835 Rectangular array: (row n) = b**c, where b(h) = h, c(h) = 4*n-7+4*h, n>=1, h>=1, and ** = convolution.

%I A213835 #11 May 25 2018 04:23:45
%S A213835 1,7,5,22,19,9,50,46,31,13,95,90,70,43,17,161,155,130,94,55,21,252,
%T A213835 245,215,170,118,67,25,372,364,329,275,210,142,79,29,525,516,476,413,
%U A213835 335,250,166,91,33,715,705,660,588,497,395
%N A213835 Rectangular array:  (row n) = b**c, where b(h) = h, c(h) = 4*n-7+4*h, n>=1, h>=1, and ** = convolution.
%C A213835 Principal diagonal: A172078.
%C A213835 Antidiagonal sums: A051797.
%C A213835 Row 1, (1,2,3,4,5,...)**(1,5,9,13,...): A002412.
%C A213835 Row 2, (1,2,3,4,5,...)**(5,9,13,17,...): (4*k^3 + 15*k^2 - 11*k)/6.
%C A213835 Row 3, (1,2,3,4,5,...)**(9,13,17,21,...): (4*k^3 + 27*k^2 - 23*k)/6
%C A213835 For a guide to related arrays, see A212500.
%H A213835 Clark Kimberling, <a href="/A213835/b213835.txt">Antidiagonals n = 1..60, flattened.</a>
%F A213835 T(n,k) = 4*T(n,k-1)-6*T(n,k-2)+4*T(n,k-3)-T(n,k-4).
%F A213835 G.f. for row n: f(x)/g(x), where f(x) = x*((4*n-3) + (4*n-7)*x) and g(x) = (1-x)^4.
%e A213835 Northwest corner (the array is read by falling antidiagonals):
%e A213835 1....7....22....50....95
%e A213835 5....19...46....90....155
%e A213835 9....31...70....130...215
%e A213835 13...43...94....170...275
%e A213835 17...55...118...210...335
%e A213835 21...67...142...250...395
%t A213835 b[n_]:=n;c[n_]:=4n-3;
%t A213835 t[n_,k_]:=Sum[b[k-i]c[n+i],{i,0,k-1}]
%t A213835 TableForm[Table[t[n,k],{n,1,10},{k,1,10}]]
%t A213835 Flatten[Table[t[n-k+1,k],{n,12},{k,n,1,-1}]]
%t A213835 r[n_]:=Table[t[n,k],{k,1,60}] (* A213835 *)
%t A213835 Table[t[n,n],{n,1,40}] (* A172078 *)
%t A213835 s[n_]:=Sum[t[i,n+1-i],{i,1,n}]
%t A213835 Table[s[n],{n,1,50}] (* A051797 *)
%Y A213835 Cf. A212500.
%Y A213835 Cf. A304659 (first lower diagonal).
%K A213835 nonn,tabl,easy
%O A213835 1,2
%A A213835 _Clark Kimberling_, Jul 04 2012