This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A213568 #22 Sep 04 2023 12:45:49
%S A213568 1,4,2,11,7,3,26,18,10,4,57,41,25,13,5,120,88,56,32,16,6,247,183,119,
%T A213568 71,39,19,7,502,374,246,150,86,46,22,8,1013,757,501,309,181,101,53,25,
%U A213568 9,2036,1524,1012,628,372,212,116,60,28,10,4083,3059,2035,1267
%N A213568 Rectangular array: (row n) = b**c, where b(h) = 2^(h-1), c(h) = n-1+h, n>=1, h>=1, and ** = convolution.
%C A213568 Principal diagonal: A213569
%C A213568 Antidiagonal sums: A047520
%C A213568 Row 1, (1,3,6,...)**(1,4,9,...): A125128
%C A213568 Row 2, (1,3,6,...)**(4,9,16,...): A095151
%C A213568 Row 3, (1,3,6,...)**(9,16,25,...): A000247
%C A213568 Row 4, (1,3,6,...)**(16,25,36...): A208638 (?)
%C A213568 For a guide to related arrays, see A213500.
%H A213568 Clark Kimberling, <a href="/A213568/b213568.txt">Antidiagonals n = 1..60, flattened</a>
%F A213568 T(n,k) = 4*T(n,k-1) - 5*T(n,k-2) + 2*T(n,k-3). - corrected by _Clark Kimberling_, Sep 03 2023
%F A213568 G.f. for row n: f(x)/g(x), where f(x) = n - (n - 1)*x and g(x) = (1 - 2*x)*(1 - x)^2.
%F A213568 T(n,k) = 2^k*(n + 1) - (n + k + 1). - _G. C. Greubel_, Jul 26 2019
%e A213568 Northwest corner (the array is read by falling antidiagonals):
%e A213568 1...4....11...26....57....120
%e A213568 2...7....18...41....88....183
%e A213568 3...10...25...56....119...246
%e A213568 4...13...32...71....150...309
%e A213568 5...16...39...86....181...372
%e A213568 6...19...46...101...212...435
%t A213568 (* First program *)
%t A213568 b[n_]:= 2^(n-1); c[n_]:= n;
%t A213568 t[n_, k_]:= Sum[b[k-i] c[n+i], {i, 0, k-1}]
%t A213568 TableForm[Table[t[n, k], {n, 1, 10}, {k, 1, 10}]]
%t A213568 Flatten[Table[t[n-k+1, k], {n, 12}, {k, n, 1, -1}]]
%t A213568 r[n_]:= Table[t[n, k], {k, 1, 60}] (* A213568 *)
%t A213568 d = Table[t[n, n], {n, 1, 40}] (* A213569 *)
%t A213568 s[n_]:= Sum[t[i, n+1-i], {i, 1, n}]
%t A213568 s1 = Table[s[n], {n, 1, 50}] (* A047520 *)
%t A213568 (* Second program *)
%t A213568 Table[2^(n-k+1)*(k+1) -(n+2), {n, 12}, {k, n}]//Flatten (* _G. C. Greubel_, Jul 26 2019 *)
%o A213568 (PARI) for(n=1,12, for(k=1,n, print1(2^(n-k+1)*(k+1) -(n+2), ", "))) \\ _G. C. Greubel_, Jul 26 2019
%o A213568 (Magma) [2^(n-k+1)*(k+1) -(n+2): k in [1..n], n in [1..12]]; // _G. C. Greubel_, Jul 26 2019
%o A213568 (Sage) [[2^(n-k+1)*(k+1) -(n+2) for k in (1..n)] for n in (1..12)] # _G. C. Greubel_, Jul 26 2019
%o A213568 (GAP) Flat(List([1..12], n-> List([1..n], k-> 2^(n-k+1)*(k+1) -(n+2) ))); # _G. C. Greubel_, Jul 26 2019
%Y A213568 Cf. A213500.
%K A213568 nonn,tabl,easy
%O A213568 1,2
%A A213568 _Clark Kimberling_, Jun 18 2012