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A213756 Rectangular array: (row n) = b**c, where b(h) = -1 + 2^h, c(h) = 2*n - 3 + 2*h, n>=1, h>=1, and ** = convolution.

%I A213756 #12 Feb 27 2018 10:27:30
%S A213756 1,6,3,21,14,5,58,43,22,7,141,110,65,30,9,318,255,162,87,38,11,685,
%T A213756 558,369,214,109,46,13,1434,1179,798,483,266,131,54,15,2949,2438,1673,
%U A213756 1038,597,318,153,62,17,5998,4975,3442,2167,1278,711,370,175,70
%N A213756 Rectangular array:  (row n) = b**c, where b(h) = -1 + 2^h, c(h) = 2*n - 3 + 2*h, n>=1, h>=1, and ** = convolution.
%C A213756 Principal diagonal: A213757.
%C A213756 Antidiagonal sums: A213758.
%C A213756 Row 1,  (1,3,7,15,31,...)**(1,3,5,7,9,...): A047520.
%C A213756 Row 2,  (1,3,7,15,31,...)**(3,5,7,9,11,...).
%C A213756 Row 3,  (1,3,7,15,31,...)**(5,7,9,11,13,...).
%C A213756 For a guide to related arrays, see A213500.
%H A213756 Clark Kimberling, <a href="/A213756/b213756.txt">Antidiagonals n = 1..40, flattened</a>
%F A213756 T(n,k) = 5*T(n,k-1)-9*T(n,k-2)+7*T(n,k-3)-2*T(n,k-4).
%F A213756 G.f. for row n:  f(x)/g(x), where f(x) = x*(2*n - 1 - (2*n - 3)*x) and g(x) = (1 - 2*x)(1 - x )^3.
%e A213756 Northwest corner (the array is read by falling antidiagonals):
%e A213756 1....6....21....58....141...318
%e A213756 3....14...43....110...255...558
%e A213756 5....22...65....162...369...798
%e A213756 7....30...87....214...483...1038
%e A213756 9....38...109...266...597...1278
%e A213756 11...46...131...318...711...1518
%t A213756 b[n_] := -1 + 2^n; c[n_] := 2 n - 1;
%t A213756 t[n_, k_] := Sum[b[k - i] c[n + i], {i, 0, k - 1}]
%t A213756 TableForm[Table[t[n, k], {n, 1, 10}, {k, 1, 10}]]
%t A213756 Flatten[Table[t[n - k + 1, k], {n, 12}, {k, n, 1, -1}]]
%t A213756 r[n_] := Table[t[n, k], {k, 1, 60}]  (* A213756 *)
%t A213756 Table[t[n, n], {n, 1, 40}] (* A213757 *)
%t A213756 s[n_] := Sum[t[i, n + 1 - i], {i, 1, n}]
%t A213756 Table[s[n], {n, 1, 50}] (* A213758 *)
%Y A213756 Cf. A213500.
%K A213756 nonn,tabl,easy
%O A213756 1,2
%A A213756 _Clark Kimberling_, Jun 20 2012