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%I #16 Apr 11 2013 17:15:35
%S 1,8,4,34,25,9,104,88,52,16,259,234,170,89,25,560,524,424,280,136,36,
%T 1092,1043,899,674,418,193,49,1968,1904,1708,1384,984,584,260,64,3333,
%U 3252,2996,2555,1979,1354,778,337,81,5368,5268,4944,4368,3584
%N Rectangular array: (row n) = b**c, where b(h) = h^2, c(h) = (n-1+h)^2, n>=1, h>=1, and ** = convolution.
%C Principal diagonal: A213546.
%C Antidiagonal sums: A213547.
%C Row 1, (1,4,9,...)**(1,4,9,...): A033455.
%C Row 2, (1,4,9,...)**(4,9,16,...): (k^5 + 10*k^4 + 40*k^3 + 50*k^2 +19*k)/30.
%C Row 3, (1,4,9,...)**(9,16,25,...): (k^5 + 15*k^4 + 90*k^3 + 120*k^2+44*k)/30.
%C For a guide to related arrays, see A213500.
%H Clark Kimberling, <a href="/A213505/b213505.txt">Antidiagonals n = 1..60, flattened</a>
%H Henri Muehle, <a href="http://arxiv.org/abs/1301.1654">Proper Mergings of Stars and Chains are Counted by Sums of Antidiagonals in Certain Convolution Arrays -- The Details</a>, arXiv preprint arXiv:1301.1654, 2013.
%F 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 G.f. for row n: f(x)/g(x), where f(x) = n^2 - (n^2 - 2*n - 1)*x - (n^2 - 2)*x^2 - ((n - 1)^2)*x^3 and g(x) = (1 - x)^6.
%e Northwest corner (the array is read by falling antidiagonals):
%e 1....8.....34....104...259....560
%e 4....25....88....234...524....1043
%e 9....52....170...424...899....1708
%e 16...89....280...674...1384...2555
%e 25...136...418...984...1979...3584
%e ...
%e T(5,1) = (1)**(25) = 25
%e T(5,2) = (1,4)**(25,36) = 1*36+4*25 = 136
%e T(5,3) = (1,4,9)**(25,36,49) = 1*49+4*36+9*25 = 418
%t b[n_] := n^2; c[n_] := n^2
%t t[n_, k_] := Sum[b[k - i] c[n + i], {i, 0, k - 1}]
%t TableForm[Table[t[n, k], {n, 1, 10}, {k, 1, 10}]]
%t Flatten[Table[t[n - k + 1, k], {n, 12}, {k, n, 1, -1}]]
%t r[n_] := Table[t[n, k], {k, 1, 60}] (* A213505 *)
%t d = Table[t[n, n], {n, 1, 40}] (* A213546 *)
%t s[n_] := Sum[t[i, n + 1 - i], {i, 1, n}]
%t s1 = Table[s[n], {n, 1, 50}] (* A213547 *)
%Y Cf. A213500.
%K nonn,tabl,easy
%O 1,2
%A _Clark Kimberling_, Jun 16 2012
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