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%I #17 Sep 08 2022 08:46:02
%S 1,6,4,21,17,9,58,50,34,16,141,125,93,57,25,318,286,222,150,86,36,685,
%T 621,493,349,221,121,49,1434,1306,1050,762,506,306,162,64,2949,2693,
%U 2181,1605,1093,693,405,209,81,5998,5486,4462,3310,2286,1486
%N Rectangular array: (row n) = b**c, where b(h) = 2^(h-1), c(h) = (n-1+h)^2, n>=1, h>=1, and ** = convolution.
%C Principal diagonal: A213574.
%C Antidiagonal sums: A213575.
%C row 1, (1,2,4,8,...)**(1,4,9,16...): A047520.
%C row 2, (1,2,4,8,...)**(4,9,16,25...).
%C row 3, (1,2,4,8,...)**(9,16,25,36...).
%C For a guide to related arrays, see A213500.
%H Clark Kimberling, <a href="/A213573/b213573.txt">Antidiagonals n = 1..60, flattened</a>
%F T(n,k) = 5*T(n,k-1) - 9*T(n,k-2) + 7*T(n,k-3) - 2*T(n,k-4).
%F G.f. for row n: f(x)/g(x), where f(x) = n^2 - (2*n^2 - 2*n - 1)*x + (n - 1)*x^2 and g(x) = (1 - 2*x)*(1 - x)^3.
%F T(n,k) = 2^k*(n^2 + 2*n + 3) - (n + k + 2)^2 + 2*(n + k + 1) - 1. - _G. C. Greubel_, Jul 25 2019
%e Northwest corner (the array is read by falling antidiagonals):
%e 1, 6, 21, 58, 141, 318, ...
%e 4, 17, 50, 125, 286, 621, ...
%e 9, 34, 93, 222, 493, 1050, ...
%e 16, 57, 150, 349, 762, 1605, ...
%e 25, 86, 221, 506, 1093, 2286, ...
%e 36, 121, 306, 693, 1486, 3093, ...
%e ...
%t (* First program *)
%t b[n_]:= 2^(n-1); c[n_]:= n^2;
%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, 60}] (* A213573 *)
%t d = Table[T[n, n], {n, 40}] (* A213574 *)
%t s[n_]:= Sum[T[i, n+1-i], {i, 1, n}]
%t s1 = Table[s[n], {n, 1, 50}] (* A213575 *)
%t (* Additional programs *)
%t Table[2^(n-k+1)*((k+1)^2 +2)-((n+2)^2 +2), {n,12}, {k, n}]//Flatten (* _G. C. Greubel_, Jul 25 2019 *)
%o (PARI) for(n=1,12, for(k=1,n, print1(2^(n-k+1)*((k+1)^2 +2)-((n+2)^2 +2), ", "))) \\ _G. C. Greubel_, Jul 25 2019
%o (Magma) [2^(n-k+1)*((k+1)^2 +2)-((n+2)^2 +2): k in [1..n], n in [1..12]]; // _G. C. Greubel_, Jul 25 2019
%o (Sage) [[2^(n-k+1)*((k+1)^2 +2)-((n+2)^2 +2) for k in (1..n)] for n in (1..12)] # _G. C. Greubel_, Jul 25 2019
%o (GAP) Flat(List([1..12], n-> List([1..n], k-> 2^(n-k+1)*((k+1)^2 +2)- ((n+2)^2 +2) ))); # _G. C. Greubel_, Jul 25 2019
%Y Cf. A213500.
%K nonn,tabl,easy
%O 1,2
%A _Clark Kimberling_, Jun 18 2012
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