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A213564 Rectangular array: (row n) = b**c, where b(h) = h*(h+1)/2, c(h) = (n-1+h)^2, n>=1, h>=1, and ** = convolution.

%I A213564 #9 Jul 12 2012 12:23:20
%S A213564 1,7,4,27,21,9,77,67,43,16,182,167,127,73,25,378,357,297,207,111,36,
%T A213564 714,686,602,467,307,157,49,1254,1218,1106,917,677,427,211,64,2079,
%U A213564 2034,1890,1638,1302,927,567,273,81,3289,3234,3054,2730,2282,1757
%N A213564 Rectangular array:  (row n) = b**c, where b(h) = h*(h+1)/2, c(h) = (n-1+h)^2, n>=1, h>=1, and ** = convolution.
%C A213564 Principal diagonal:  A213565
%C A213564 Antidiagonal sums:  A101094
%C A213564 Row 1,  (1,3,6,...)**(1,4,9,...):  A005585
%C A213564 Row 2,  (1,3,6,...)**(4,9,16,...):  (k^5 +25*k^4 + 60*k^3 + 215*k^2 + 59*k)/60
%C A213564 Row 3,  (1,3,6,...)**(9,16,25,...):  (k^5 +35*k^4 + 30*k^3 + 505*k^2 + 149*k)/60
%C A213564 For a guide to related arrays, see A213500.
%H A213564 Clark Kimberling, <a href="/A213564/b213564.txt">Antidiagonals n = 1..60, flattened</a>
%F A213564 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 A213564 G.f. for row n:  f(x)/g(x), where f(x) = n^2 - (2*n^2 - 2n - 1)*x + ((n - 1)^2)*x^2 and g(x) = (1 - x)^6.
%e A213564 Northwest corner (the array is read by falling antidiagonals):
%e A213564 1....7.....27....77....182
%e A213564 4....21....67....167...357
%e A213564 9....43....127...297...602
%e A213564 16...73....207...467...917
%e A213564 25...111...307...677...1302
%e A213564 36...157...427...927...1757
%t A213564 b[n_] := n (n + 1)/2; c[n_] := n^2
%t A213564 t[n_, k_] := Sum[b[k - i] c[n + i], {i, 0, k - 1}]
%t A213564 TableForm[Table[t[n, k], {n, 1, 10}, {k, 1, 10}]]
%t A213564 Flatten[Table[t[n - k + 1, k], {n, 12}, {k, n, 1, -1}]]
%t A213564 r[n_] := Table[t[n, k], {k, 1, 60}]  (* A213564 *)
%t A213564 d = Table[t[n, n], {n, 1, 40}] (* A213565 *)
%t A213564 s[n_] := Sum[t[i, n + 1 - i], {i, 1, n}]
%t A213564 s1 = Table[s[n], {n, 1, 50}] (* A101094 *)
%Y A213564 Cf. A213500.
%K A213564 nonn,tabl,easy
%O A213564 1,2
%A A213564 _Clark Kimberling_, Jun 18 2012