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 A213571 #21 Sep 08 2022 08:46:02
%S A213571 1,5,3,16,13,7,42,38,29,15,99,94,82,61,31,219,213,198,170,125,63,466,
%T A213571 459,441,406,346,253,127,968,960,939,897,822,698,509,255,1981,1972,
%U A213571 1948,1899,1809,1654,1402,1021,511,4017,4007,3980,3924,3819,3633
%N A213571 Rectangular array: (row n) = b**c, where b(h) = h, c(h) = (n-1+h)^2, n>=1, h>=1, and ** = convolution.
%C A213571 Principal diagonal: A213572.
%C A213571 Antidiagonal sums: A213581.
%C A213571 Row 1, (1,2,3,4,5,...)**(1,3,7,15,31,...): A002662.
%C A213571 Row 2, (1,2,3,4,5,...)**(3,7,15,31,63,...).
%C A213571 Row 3, (1,2,3,4,5,...)**(7,15,31,63,...).
%C A213571 For a guide to related arrays, see A213500.
%H A213571 Clark Kimberling, <a href="/A213571/b213571.txt">antidiagonals n = 1..60, flattened</a>
%F A213571 T(n,k) = 5*T(n,k-1) - 9*T(n,k-2) + 7*T(n,k-3) - 2*T(n,k-4).
%F A213571 G.f. for row n: f(x)/g(x), where f(x) = x*(-1 + 2^n - (-2 + 2^n)*x) and g(x) = (1 - 2*x)(1 - x)^3.
%F A213571 T(n,k) = 2^(n+k+1) - 2^n*(k+2) - binomial(k+1, 2). - _G. C. Greubel_, Jul 25 2019
%e A213571 Northwest corner (the array is read by falling antidiagonals):
%e A213571 1, 5, 16, 42, 99, 219, ...
%e A213571 3, 13, 38, 94, 213, 459, ...
%e A213571 7, 29, 82, 198, 441, 939, ...
%e A213571 15, 61, 170, 406, 897, 1899, ...
%e A213571 31, 125, 346, 822, 1809, 3819, ...
%e A213571 ...
%t A213571 (* First program *)
%t A213571 b[n_]:= n; c[n_]:= -1 + 2^n;
%t A213571 t[n_, k_]:= Sum[b[k-i] c[n+i], {i, 0, k-1}]
%t A213571 TableForm[Table[t[n, k], {n, 1, 10}, {k, 1, 10}]]
%t A213571 Flatten[Table[t[n-k+1, k], {n, 12}, {k, n, 1, -1}]]
%t A213571 r[n_]:= Table[t[n, k], {k, 1, 60}] (* A213571 *)
%t A213571 d = Table[t[n, n], {n, 1, 40}] (* A213572 *)
%t A213571 s[n_]:= Sum[t[i, n+1-i], {i, 1, n}]
%t A213571 s1 = Table[s[n], {n, 1, 50}] (* A213581 *)
%t A213571 (* Additional programs *)
%t A213571 Table[2^(n+2) -2^k*(n-k+3) -Binomial[n-k+2, 2], {n,12}, {k, n}]//Flatten (* _G. C. Greubel_, Jul 25 2019 *)
%o A213571 (PARI) for(n=1,12, for(k=1,n, print1(2^(n+2) -2^k*(n-k+3) -binomial(n-k+2, 2), ", "))) \\ _G. C. Greubel_, Jul 25 2019
%o A213571 (Magma) [2^(n+2) -2^k*(n-k+3) -Binomial(n-k+2, 2): k in [1..n], n in [1..12]]; // _G. C. Greubel_, Jul 25 2019
%o A213571 (Sage) [[2^(n+2) -2^k*(n-k+3) -binomial(n-k+2, 2) for k in (1..n)] for n in (1..12)] # _G. C. Greubel_, Jul 25 2019
%o A213571 (GAP) Flat(List([1..12], n-> List([1..n], k-> 2^(n+2) -2^k*(n-k+3) -Binomial(n-k+2, 2) ))); # _G. C. Greubel_, Jul 25 2019
%Y A213571 Cf. A213500, A213587.
%K A213571 nonn,tabl,easy
%O A213571 1,2
%A A213571 _Clark Kimberling_, Jun 19 2012