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 A303929 #11 Jun 14 2018 04:03:50
%S A303929 1,1,1,1,1,1,1,1,1,1,1,1,1,2,1,1,1,1,2,3,1,1,1,1,3,5,6,1,1,1,1,3,8,13,
%T A303929 12,1,1,1,1,4,11,34,49,27,1,1,1,1,4,16,60,169,201,65,1,1,1,1,5,20,109,
%U A303929 423,1019,940,175,1,1,1,1,5,26,167,918,3381,6710,4643,490,1
%N A303929 Array read by antidiagonals: T(n,k) is the number of noncrossing partitions up to rotation and reflection composed of n blocks of size k.
%H A303929 Andrew Howroyd, <a href="/A303929/b303929.txt">Table of n, a(n) for n = 0..1274</a>
%H A303929 Wikipedia, <a href="https://en.wikipedia.org/wiki/Noncrossing_partition">Noncrossing partition</a>
%e A303929 =================================================================
%e A303929 n\k| 1 2 3 4 5 6 7 8 9
%e A303929 ---+-------------------------------------------------------------
%e A303929 0 | 1 1 1 1 1 1 1 1 1 ...
%e A303929 1 | 1 1 1 1 1 1 1 1 1 ...
%e A303929 2 | 1 1 1 1 1 1 1 1 1 ...
%e A303929 3 | 1 2 2 3 3 4 4 5 5 ...
%e A303929 4 | 1 3 5 8 11 16 20 26 32 ...
%e A303929 5 | 1 6 13 34 60 109 167 257 359 ...
%e A303929 6 | 1 12 49 169 423 918 1741 3051 4969 ...
%e A303929 7 | 1 27 201 1019 3381 9088 20569 41769 77427 ...
%e A303929 8 | 1 65 940 6710 29335 96315 259431 607696 1280045 ...
%e A303929 9 | 1 175 4643 47104 266703 1072187 3417520 9240444 22066742 ...
%e A303929 ...
%t A303929 u[n_, k_, r_] := (r*Binomial[k*n + r, n]/(k*n + r));
%t A303929 e[n_, k_] := Sum[ u[j, k, 1 + (n - 2*j)*k/2], {j, 0, n/2}]
%t A303929 c[n_, k_] := If[n == 0, 1, (DivisorSum[n, EulerPhi[n/#]*Binomial[k*#, #]&] + DivisorSum[GCD[n - 1, k], EulerPhi[#]*Binomial[n*k/#, (n - 1)/#]&])/(k*n) - Binomial[k*n, n]/(n*(k - 1) + 1)];
%t A303929 T[n_, k_] := (1/2)*(c[n, k] + If[n == 0, 1, If[OddQ[k], If[OddQ[n], 2*u[ Quotient[n, 2], k, (k + 1)/2], u[n/2, k, 1] + u[n/2 - 1, k, k]], e[n, k] + If[OddQ[n], u[Quotient[n, 2], k, k/2]]]/2]) /. Null -> 0;
%t A303929 Table[T[n - k, k], {n, 1, 12}, {k, n, 1, -1}] // Flatten (* _Jean-François Alcover_, Jun 14 2018, translated from PARI *)
%o A303929 (PARI) \\ here c(n,k) is A303694
%o A303929 u(n,k,r) = {r*binomial(k*n + r, n)/(k*n + r)}
%o A303929 e(n,k) = {sum(j=0, n\2, u(j, k, 1+(n-2*j)*k/2))}
%o A303929 c(n, k)={if(n==0, 1, (sumdiv(n, d, eulerphi(n/d)*binomial(k*d, d)) + sumdiv(gcd(n-1, k), d, eulerphi(d)*binomial(n*k/d, (n-1)/d)))/(k*n) - binomial(k*n, n)/(n*(k-1)+1))}
%o A303929 T(n,k)={(1/2)*(c(n,k) + if(n==0, 1, if(k%2, if(n%2, 2*u(n\2,k,(k+1)/2), u(n/2,k,1) + u(n/2-1,k,k)), e(n,k) + if(n%2, u(n\2,k,k/2)))/2))}
%Y A303929 Columns 2..5 are A006082(n+1), A082938, A303870, A303871.
%Y A303929 Cf. A111275, A209612, A211359, A303694, A303875.
%K A303929 nonn,tabl
%O A303929 0,14
%A A303929 _Andrew Howroyd_, May 02 2018