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A213781 Rectangular array: (row n) = b**c, where b(h) = 1+[h/2], c(h) = n-1+h, n>=1, h>=1, [ ] = floor, and ** = convolution.

%I A213781 #11 Jul 12 2012 12:17:36
%S A213781 1,4,2,9,7,3,17,14,10,4,28,25,19,13,5,43,39,33,24,16,6,62,58,50,41,29,
%T A213781 19,7,86,81,73,61,49,34,22,8,115,110,100,88,72,57,39,25,9,150,144,134,
%U A213781 119,103,83,65,44,28,10,191,185,173,158,138,118,94,73,49,31
%N A213781 Rectangular array:  (row n) = b**c, where b(h) = 1+[h/2], c(h) = n-1+h, n>=1, h>=1, [ ] = floor, and ** = convolution.
%C A213781 Principal diagonal: A213782.
%C A213781 Antidiagonal sums: A005712.
%C A213781 row 1,  (1,2,2,3,3,4,4,...)**(1,2,3,4,5,6,7,...): A005744.
%C A213781 row 2,  (1,2,2,3,3,4,4,...)**(2,3,4,5,6,7,8,...).
%C A213781 row 3,  (1,2,2,3,3,4,4,...)**(3,4,5,6,7,8,9,...).
%C A213781 For a guide to related arrays, see A213500.
%H A213781 Clark Kimberling, <a href="/A213781/b213781.txt">Antidiagonals n = 1..60, flattened</a>
%F A213781 T(n,k) = 3*T(n,k-1) - 2*T(n,k-2) - 2*T(n,k-3) + 3*T(n,k-4) - T(n,k-5).
%F A213781 G.f. for row n:  f(x)/g(x), where f(x) = x*(n + x - (2*n - 1)*x^2 + (n -1)*x^3) and g(x) = (1 + x)(1 - x)^4.
%e A213781 Northwest corner (the array is read by falling antidiagonals):
%e A213781 1...4....9....17...28...43....62
%e A213781 2...7....14...25...39...58....81
%e A213781 3...10...19...33...50...73....100
%e A213781 4...13...24...41...61...88....119
%e A213781 5...16...29...49...72...103...138
%e A213781 6...19...34...57...83...118...157
%e A213781 7...22...39...65...94...133...176
%t A213781 b[n_] := Floor[(n + 2)/2]; c[n_] := n;
%t A213781 t[n_, k_] := Sum[b[k - i] c[n + i], {i, 0, k - 1}]
%t A213781 TableForm[Table[t[n, k], {n, 1, 10}, {k, 1, 10}]]
%t A213781 Flatten[Table[t[n - k + 1, k], {n, 12}, {k, n, 1, -1}]]
%t A213781 r[n_] := Table[t[n, k], {k, 1, 60}]  (* A213781 *)
%t A213781 s[n_] := Sum[t[i, n + 1 - i], {i, 1, n}]
%t A213781 s1 = Table[s[n], {n, 1, 50}] (* A005712 *)
%Y A213781 Cf. A213500, A213778.
%K A213781 nonn,tabl,easy
%O A213781 1,2
%A A213781 _Clark Kimberling_, Jun 22 2012