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%I #4 Jun 22 2012 13:46:14
%S 1,6,3,19,14,5,44,37,22,7,85,76,55,30,9,146,135,108,73,38,11,231,218,
%T 185,140,91,46,13,344,329,290,235,172,109,54,15,489,472,427,362,285,
%U 204,127,62,17,670,651,600,525,434,335,236,145,70,19,891,870,813
%N Rectangular array: (row n) = b**c, where b(h) = 2*h-1, c(h) = b(n-1+h), n>=1, h>=1, and ** = convolution.
%C Principal diagonal: A100157
%C Antidiagonal sums: A071238
%C row 1, (1,3,5,7,9,...)**(1,3,5,7,9,...): A005900
%C row 2, (1,3,5,7,9,...)**(3,5,7,9,11,...): A143941
%C row 3, (1,3,5,7,9,...)**(5,7,9,11,13,...): (2*k^3 + 12*k^2 + k)/6
%C row 4, (1,3,5,7,9,...)**(7,9,11,13,15,,...): (2*k^3 + 18*k^2 + k)/6
%C For a guide to related arrays, see A213500.
%F T(n,k) = 4*T(n,k-1)-6*T(n,k-2)+4*T(n,k-3)-T(n,k-4).
%F G.f. for row n: f(x)/g(x), where f(x) = 2*n - 1 + 2*x - (2*n - 3)*x^2 and g(x) = (1 - x )^4.
%e Northwest corner (the array is read by falling antidiagonals):
%e 1...6....19...44....85....146
%e 3...14...37...76....135...218
%e 5...22...55...108...185...290
%e 7...30...73...140...235...362
%e 9...38...91...172...285...434
%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}] (* A213752 *)
%t Table[t[n, n], {n, 1, 40}] (* A100157 *)
%t s[n_] := Sum[t[i, n + 1 - i], {i, 1, n}]
%t Table[s[n], {n, 1, 50}] (* A071238 *)
%Y Cf. A213500.
%K nonn,tabl,easy
%O 1,2
%A _Clark Kimberling_, Jun 20 2012
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