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%I #12 Jul 11 2012 08:51:12
%S 1,6,3,21,16,7,58,51,36,15,141,132,111,76,31,318,307,280,231,156,63,
%T 685,672,639,576,471,316,127,1434,1419,1380,1303,1168,951,636,255,
%U 2949,2932,2887,2796,2631,2352,1911,1276,511,5998,5979,5928,5823
%N Rectangular array: (row n) = b**c, where b(h) = 2*h-1, c(h) = -1 + 2^(n-1+h), n>=1, h>=1, and ** = convolution.
%C Principal diagonal: A213754.
%C Antidiagonal sums: A213755.
%C Row 1, (1,3,5,7,9,...)**(1,3,7,15,...): A047520.
%C Row 2, (1,3,5,7,9,...)**(3,7,15,31,...).
%C Row 3, (1,3,5,7,9,...)**(7,15,31,63...).
%C Ror a guide to related arrays, see A213500.
%H Clark Kimberling, <a href="/A213753/b213753.txt">Antidiagonals n = 1..60, flattened</a>
%F T(n,k) = 5*T(n,k-1)-9*T(n,k-2)+7*T(n,k-3)-2*T(n,k-4).
%F G.f. for row n: f(x)/g(x), where f(x) = x*(-1 + 2^n + x + (-2 + 2^n)*x^2) and g(x) = (1 - 2*x)(1 - x )^3.
%e Northwest corner (the array is read by falling antidiagonals):
%e 1....6.....21....58.....141
%e 3....16....51....132....307
%e 7....36....111...280....639
%e 15...76....231...576....1303
%e 31...156...471...1168...2631
%t b[n_] := 2 n - 1; c[n_] := -1 + 2^n;
%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}] (* A213753 *)
%t Table[t[n, n], {n, 1, 40}] (* A213754 *)
%t s[n_] := Sum[t[i, n + 1 - i], {i, 1, n}]
%t Table[s[n], {n, 1, 50}] (* A213755 *)
%Y Cf. A213500.
%K nonn,tabl,easy
%O 1,2
%A _Clark Kimberling_, Jun 20 2012
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