Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).
%I #21 Aug 02 2023 02:00:40
%S 1,16,243,4112,78125,1680345,40353607,1073766400,31381060338,
%T 1000000781250,34522712143931,1283918489808640,51185893014090757,
%U 2177953338656796883,98526125335697265625,4722366482899710050304,239072435685151324847153
%N a(n) = Sum_{d|n} d^(d+n/d) * binomial(d,n/d).
%H Winston de Greef, <a href="/A360759/b360759.txt">Table of n, a(n) for n = 1..383</a>
%F G.f.: Sum_{k>0} k^k * ( (1 + k*x^k)^k - 1 ).
%F If p is prime, a(p) = p^(p+2).
%t a[n_] := DivisorSum[n, #^(# + n/#) * Binomial[#, n/#] &]; Array[a, 20] (* _Amiram Eldar_, Aug 02 2023 *)
%o (PARI) a(n) = sumdiv(n, d, d^(d+n/d)*binomial(d, n/d));
%o (PARI) my(N=20, x='x+O('x^N)); Vec(sum(k=1, N, k^k*((1+k*x^k)^k-1)))
%Y Cf. A318636, A327238, A338685, A338693, A338694.
%Y Cf. A360712.
%K nonn
%O 1,2
%A _Seiichi Manyama_, Feb 19 2023