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A213555 Rectangular array: (row n) = b**c, where b(h) = h^3, c(h) = n-1+h, n>=1, h>=1, and ** = convolution.

%I A213555 #13 Jul 11 2012 18:47:20
%S A213555 1,10,2,46,19,3,146,82,28,4,371,246,118,37,5,812,596,346,154,46,6,
%T A213555 1596,1253,821,446,190,55,7,2892,2380,1694,1046,546,226,64,8,4917,
%U A213555 4188,3164,2135,1271,646,262,73,9,7942,6942,5484,3948,2576,1496,746
%N A213555 Rectangular array:  (row n) = b**c, where b(h) = h^3, c(h) = n-1+h, n>=1, h>=1, and ** = convolution.
%C A213555 Principal diagonal: A213556.
%C A213555 Antidiagonal sums: A213547.
%C A213555 Row 1,  (1,8,27,...)**(1,2,3,...):  A024166.
%C A213555 Row 2,  (1,8,27,...)**(2,3,4,...): (3*k^5 + 30*k^4 + 55*k^3 + 30*k^2 + 2*k)/60.
%C A213555 Row 3,  (1,8,27,...)**(3,4,5,...): (3*k^5 + 45*k^4 + 85*k^3 + 45*k^2 + 2*k)/60.
%C A213555 For a guide to related arrays, see A213500.
%H A213555 Clark Kimberling, <a href="/A213555/b213555.txt">Antidiagonals n = 1..60, flattened</a>
%F A213555 T(n,k) = 6*T(n,k-1) - 15*T(n,k-2) + 20*T(n,k-3) - 15*T(n,k-4) + 6*T(n,k-5) -T(n,k-6).
%F A213555 G.f. for row n:  f(x)/g(x), where f(x) = n + (3*n + 1)*x - (3*n - 4)*x^2 - (n - 1)*x^3 and g(x) = (1 - x)^6.
%e A213555 Northwest corner (the array is read by falling antidiagonals):
%e A213555 1...10...46....146...371....812
%e A213555 2...19...82....246...596....1253
%e A213555 3...28...118...346...821....1694
%e A213555 4...37...154...446...1046...2135
%e A213555 5...46...190...546...1271...2576
%e A213555 6...55...226...646...1496...3017
%t A213555 b[n_] := n^3; c[n_] := n
%t A213555 t[n_, k_] := Sum[b[k - i] c[n + i], {i, 0, k - 1}]
%t A213555 TableForm[Table[t[n, k], {n, 1, 10}, {k, 1, 10}]]
%t A213555 Flatten[Table[t[n - k + 1, k], {n, 12}, {k, n, 1, -1}]]
%t A213555 r[n_] := Table[t[n, k], {k, 1, 60}]  (* A213555 *)
%t A213555 d = Table[t[n, n], {n, 1, 40}] (* A213556 *)
%t A213555 s[n_] := Sum[t[i, n + 1 - i], {i, 1, n}]
%t A213555 s1 = Table[s[n], {n, 1, 50}] (* A213547 *)
%Y A213555 Cf. A213500, A213553.
%K A213555 nonn,tabl,easy
%O A213555 1,2
%A A213555 _Clark Kimberling_, Jun 17 2012