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A213778 Rectangular array: (row n) = b**c, where b(h) = h, c(h) = 1+[(n-1+h)/2], n>=1, h>=1, [ ] = floor, and ** = convolution.

%I A213778 #12 Jul 12 2012 12:21:41
%S A213778 1,4,2,9,6,2,17,13,7,3,28,23,15,9,3,43,37,27,19,10,4,62,55,43,33,21,
%T A213778 12,4,86,78,64,52,37,25,13,5,115,106,90,76,58,43,27,15,5,150,140,122,
%U A213778 106,85,67,47,31,16,6,191,180,160,142,118,97,73,53,33,18,6,239
%N A213778 Rectangular array:  (row n) = b**c, where b(h) = h, c(h) = 1+[(n-1+h)/2], n>=1, h>=1, [ ] = floor, and ** = convolution.
%C A213778 Principal diagonal: A213779.
%C A213778 Antidiagonal sums: A213780.
%C A213778 Row 1,  (1,2,3,4,5,...)**(1,2,2,3,3,4,4,...): A005744.
%C A213778 Row 2,  (1,2,3,4,5,...)**(2,2,3,3,4,4,...)
%C A213778 Row 3,  (1,2,3,4,5,...)**(3,4,4,5,5,...)
%C A213778 For a guide to related arrays, see A213500.
%H A213778 Clark Kimberling, <a href="/A213778/b213778.txt">Antidiagonals n=1..80, flattened</a>
%F A213778 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 A213778 G.f. for row n:  f(x)/g(x), where f(x) = x*(1 + [n/2] + d(n)*x - [(n+1)/2]*x^2), g(x) = (1 + x)*(1 - x)^4, d(n) = (n mod 2) and [] = floor.
%e A213778 Northwest corner (the array is read by falling antidiagonals):
%e A213778 1...4....9....17...28...43....62
%e A213778 2...6....13...23...37...55....78
%e A213778 2...7....15...27...43...64....90
%e A213778 3...9....19...33...52...76....106
%e A213778 3...10...21...37...58...85....118
%e A213778 4...12...25...43...67...97....134
%e A213778 4...13...27...47...73...106...146
%t A213778 b[n_] := n; c[n_] := 1 + Floor[n/2];
%t A213778 t[n_, k_] := Sum[b[k - i] c[n + i], {i, 0, k - 1}]
%t A213778 TableForm[Table[t[n, k], {n, 1, 10}, {k, 1, 10}]]
%t A213778 Flatten[Table[t[n - k + 1, k], {n, 12}, {k, n, 1, -1}]]
%t A213778 r[n_] := Table[t[n, k], {k, 1, 60}]  (* A213778 *)
%t A213778 Table[t[n, n], {n, 1, 40}] (* A213779 *)
%t A213778 s[n_] := Sum[t[i, n + 1 - i], {i, 1, n}]
%t A213778 Table[s[n], {n, 1, 50}] (* A213780 *)
%Y A213778 Cf. A213500.
%K A213778 nonn,tabl,easy
%O A213778 1,2
%A A213778 _Clark Kimberling_, Jun 21 2012