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A213751 Rectangular array: (row n) = b**c, where b(h) = 2*h-1, c(h) = n-1+h, n>=1, h>=1, and ** = convolution.

%I A213751 #4 Jun 22 2012 13:46:28
%S A213751 1,5,2,14,9,3,30,23,13,4,55,46,32,17,5,91,80,62,41,21,6,140,127,105,
%T A213751 78,50,25,7,204,189,163,130,94,59,29,8,285,268,238,199,155,110,68,33,
%U A213751 9,385,366,332,287,235,180,126,77,37,10,506,485,447,396,336,271
%N A213751 Rectangular array:  (row n) = b**c, where b(h) = 2*h-1, c(h) = n-1+h, n>=1, h>=1, and ** = convolution.
%C A213751 Principal diagonal:  A051662
%C A213751 Antidiagonal sums:  A006325
%C A213751 row 1,  (1,3,5,7,9,...)**(1,2,3,4,5,...): A000330
%C A213751 row 2,  (1,3,5,7,9,...)**(2,3,4,5,6,...): A101986
%C A213751 row 3,  (1,3,5,7,9,...)**(3,4,5,6,7,...): (2*k^3 + 15*k^2 + k)/6
%C A213751 row 4,  (1,3,5,7,9,...)**(4,5,6,7,8,...): (2*k^3 + 21*k^2 + k)/6
%C A213751 For a guide to related arrays, see A213500.
%F A213751 T(n,k) = 4*T(n,k-1)-6*T(n,k-2)+4*T(n,k-3)-T(n,k-4).
%F A213751 G.f. for row n:  f(x)/g(x), where f(x) = n + x - (n - 1)*x^2 and g(x) = (1 - x )^4.
%e A213751 Northwest corner (the array is read by falling antidiagonals):
%e A213751 1...5....14...30....55....91
%e A213751 2...9....23...46....80....127
%e A213751 3...13...32...62....105...163
%e A213751 4...17...41...78....130...199
%e A213751 5...21...50...94....155...235
%e A213751 6...25...59...110...180...271
%t A213751 b[n_] := 2 n - 1; c[n_] := n;
%t A213751 t[n_, k_] := Sum[b[k - i] c[n + i], {i, 0, k - 1}]
%t A213751 TableForm[Table[t[n, k], {n, 1, 10}, {k, 1, 10}]]
%t A213751 Flatten[Table[t[n - k + 1, k], {n, 12}, {k, n, 1, -1}]]
%t A213751 r[n_] := Table[t[n, k], {k, 1, 60}]  (* A213751 *)
%t A213751 Table[t[n, n], {n, 1, 40}] (* A051662 *)
%t A213751 s[n_] := Sum[t[i, n + 1 - i], {i, 1, n}]
%t A213751 Table[s[n], {n, 1, 50}] (* A006325 *)
%Y A213751 Cf. A213500.
%K A213751 nonn,tabl,easy
%O A213751 1,2
%A A213751 _Clark Kimberling_, Jun 20 2012