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%I #15 Jul 29 2012 18:22:33
%S 1,6,3,23,16,7,72,57,36,15,201,170,125,76,31,522,459,366,261,156,63,
%T 1291,1164,975,758,533,316,127,3084,2829,2448,2007,1542,1077,636,255,
%U 7181,6670,5905,5016,4071,3110,2165,1276,511,16398,15375,13842
%N Rectangular array: (row n) = b**c, where b(h) = -1 + 2^h, c(h) = b(n-1+h), n>=1, h>=1, and ** = convolution.
%C Principal diagonal: A213748.
%C Antidiagonal sums: A213749.
%C Row 1, (1,3,7,15,31,...)**(1,3,7,15,31,...): A045618.
%C Row 2, (1,3,7,15,31,...)**(3,7,15,31,...).
%C Row 3, (1,3,7,15,31,...)**(7,15,31,...).
%C For a guide to related arrays, see A213500.
%H Clark Kimberling, <a href="/A213747/b213747.txt">Antidiagonals n = 1..60, flattened</a>
%F T(n,k) = 6*T(n,k-1)-13*T(n,k-2)+12*T(n,k-3)-4*T(n,k-4).
%F G.f. for row n: f(x)/g(x), where f(x) = -1 + 2^n - (-2 - 2^n)*x and g(x) = (1 - 3*x + 2*x^2 )^2.
%e Northwest corner (the array is read by falling antidiagonals):
%e 1....6.....23....72.....201
%e 3....16....57....170....459
%e 7....36....125...366....975
%e 15...76....261...758....1007
%e 31...156...533...1542...4071
%t b[n_] := -1 + 2^n; 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}] (* A213747 *)
%t Table[t[n, n], {n, 1, 40}] (* A213748 *)
%t s[n_] := Sum[t[i, n + 1 - i], {i, 1, n}]
%t Table[s[n], {n, 1, 50}] (* A213749 *)
%Y Cf. A213500.
%K nonn,tabl,easy
%O 1,2
%A _Clark Kimberling_, Jun 19 2012
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