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%I #4 Jun 21 2012 22:28:36
%S 1,5,3,14,11,5,30,26,17,7,55,50,38,23,9,91,85,70,50,29,11,140,133,115,
%T 90,62,35,13,204,196,175,145,110,74,41,15,285,276,252,217,175,130,86,
%U 47,17,385,375,348,308,259,205,150,98,53,19,506,495,465,420
%N Rectangular array: (row n) = b**c, where b(h) = h, c(h) = 2*(n-1+h)-1, n>=1, h>=1, and ** = convolution.
%C Principal diagonal: A007585
%C Antidiagonal sums: A002417
%C row 1, (1,2,3,4,5,...)**(1,3,5,7,9,...): A000330
%C row 2, (1,2,3,4,5,...)**(3,5,7,9,...): A051925
%C row 3, (1,2,3,4,5,...)**(5,7,9,11,...): (2*k^3 + 15*k^2 + 13*k)/6
%C row 4, (1,2,3,4,5,...)**(7,9,11,13,...): (2*k^3 + 21*k^2 + 19*k)/6
%C For a guide to related arrays, see A213500.
%F T(n,k) = 4*T(n,k-1)-6*T(n,k-2)+4*T(n,k-3)-T(n,k-4).
%F G.f. for row n: f(x)/g(x), where f(x) = (2*n - 1) - (2*n - 3)*x and g(x) = (1 - x )^4.
%e Northwest corner (the array is read by falling antidiagonals):
%e 1....5....14...30....55....91
%e 3....11...26...50....85....133
%e 5....17...38...70....115...175
%e 7....23...50...90....145...217
%e 9....29...62...110...175...259
%e 11...35...74...130...205...301
%t b[n_] := n; c[n_] := 2 n - 1;
%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}] (* A213750 *)
%t d = Table[t[n, n], {n, 1, 40}] (* A007585 *)
%t s1 = Table[s[n], {n, 1, 50}] (* A002417 *)
%t FindLinearRecurrence[s1]
%t FindGeneratingFunction[s1, x]
%Y Cf. A213500.
%K nonn,tabl,easy
%O 1,2
%A _Clark Kimberling_, Jun 20 2012
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