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 A359842 #13 Jan 19 2023 09:33:58
%S A359842 1,0,1,90,13690,3443275,1308315371,701623884514,505274768721332,
%T A359842 470638793249281593,550707386335951810915,790898932162231992184327,
%U A359842 1367864138835420575101044139,2804370191530797723173615407860,6725366044028696102055907486691290
%N A359842 a(n) = Sum_{k=0..n} binomial(n*k,n+k).
%H A359842 Seiichi Manyama, <a href="/A359842/b359842.txt">Table of n, a(n) for n = 0..205</a>
%F A359842 a(n) ~ binomial(n^2,2*n).
%F A359842 a(n) ~ exp(2*n-2) * n^(2*n - 1/2) / (sqrt(Pi) * 2^(2*n+1)).
%F A359842 From _Peter Bala_, Jan 19 2023: (Start)
%F A359842 Conjectures: a(2^k) == 0 (mod 2^(k-1)) and a(3^k) == 0 (mod 3^(k+2)) for k >= 2; a(p^k) == 0 (mod p^(k+1)) for all primes p >= 5.
%F A359842 Let m be a positive integer. Similar recurrences may hold for the sequence whose n-th term is given by Sum_{k = 0..n} binomial(m*n*k, n+k). Cf. A099237. (End)
%p A359842 a := proc (n) option remember; add(binomial(n*k, n+k), k = 0..n) end:
%p A359842 seq(a(n), n = 0..20); # _Peter Bala_, Jan 16 2023
%t A359842 Table[Sum[Binomial[n*k, n+k], {k, 0, n}], {n, 0, 20}]
%Y A359842 Cf. A096131, A099237, A226391.
%K A359842 nonn
%O A359842 0,4
%A A359842 _Vaclav Kotesovec_, Jan 15 2023