A213762 - OEIS

%I #11 Jul 11 2012 18:37:00

%S 1,5,3,15,11,5,37,29,17,7,83,67,43,23,9,177,145,97,57,29,11,367,303,

%T 207,127,71,35,13,749,621,429,269,157,85,41,15,1515,1259,875,555,331,

%U 187,99,47,17,3049,2537,1769,1129,681,393,217,113,53,19,6119

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

%C Principal diagonal: A213763.

%C Antidiagonal sums: A213764.

%C Row 1,  (1,2,4,8,16,...)**(1,3,5,7,9,...): A050488.

%C Row 2,  (1,2,4,8,16,...)**(3,5,7,9,11,...).

%C Row 3,  (1,2,4,8,16,...)**(5,7,9,11,13,...).

%C For a guide to related arrays, see A213500.

%H Clark Kimberling, <a href="/A213762/b213762.txt">Antidiagonals n = 1..80, flattened</a>

%F T(n,k) = 4*T(n,k-1)-5*T(n,k-2)+2*T(n,k-3).

%F 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 )^2.

%e Northwest corner (the array is read by falling antidiagonals):

%e 1....5....15...37....83....177

%e 3....11...29...67....145...303

%e 5....17...43...97....207...429

%e 7....23...57...127...269...555

%e 9....29...71...157...331...681

%e 11...35...85...187...393...807

%t b[n_] := 2^(n - 1); 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}]  (* A213762 *)

%t Table[t[n, n], {n, 1, 40}] (* A213763 *)

%t s[n_] := Sum[t[i, n + 1 - i], {i, 1, n}]

%t Table[s[n], {n, 1, 50}] (* A213764 *)

%Y Cf. A213500, A213756.

%K nonn,tabl,easy

%O 1,2

%A _Clark Kimberling_, Jun 20 2012