|
%I #18 Jan 19 2023 09:33:53
%S 1,1,3,10,45,251,1624,11908,97545,880660,8664546,92096731,1050304775,
%T 12778138842,165033693175,2253204163256,32401745953105,
%U 489207829112931,7733130368443057,127664099576228184,2196149923000824756
%N a(n) = Sum_{k=0..n} binomial(n*(n-k), k).
%C Main diagonal of A099233.
%H Vaclav Kotesovec, <a href="/A099237/b099237.txt">Table of n, a(n) for n = 0..490</a>
%F From _Vaclav Kotesovec_, Feb 19 2018: (Start)
%F a(n)^(1/n) ~ n^(n/w) * (n+1-w)^(1 - (n+1)/w) * (w-1)^(1/w - 1), where w = LambertW(exp(1)*n),
%F a(n)^(1/n) ~ n/log(n), but the convergence is too slow. (End)
%F From _Peter Bala_, Jan 19 2023: (Start)
%F Conjectures: a(2^k) == 1 (mod 2^k) and a(3^k) == 1 (mod 3^(k+1)); a(p^k) == 1 (mod p^(k+1)) for all primes p >= 5.
%F 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. A359842. (End)
%p A099237:= n-> add( binomial(n*j, n-j), j=0..n );
%p seq(A099237(n), n=0..30); # _G. C. Greubel_, Mar 09 2021
%t Table[Sum[Binomial[n*(n - k), k], {k, 0, n}], {n, 0, 30}] (* _Vaclav Kotesovec_, Feb 19 2018 *)
%o (Sage)
%o def A099237(n): return sum( binomial(n*j, n-j) for j in (0..n))
%o [A099237(n) for n in (0..30)] # _G. C. Greubel_, Mar 09 2021
%o (Magma)
%o A099237:= func< n | (&+[Binomial(n*j, n-j): j in [0..n]]) >;
%o [A099237(n): n in [0..30]]; // _G. C. Greubel_, Mar 09 2021
%Y Cf. A099233, A157114, A359842.
%K easy,nonn
%O 0,3
%A _Paul Barry_, Oct 08 2004
| 