%I #16 Aug 02 2023 01:59:59

%S 1,5,27,260,3125,46684,823543,16777472,387420498,10000003125,

%T 285311670611,8916100495009,302875106592253,11112006826381559,

%U 437893890380860625,18446744073726328848,827240261886336764177,39346408075296925015353

%N Expansion of Sum_{k>0} (x * (k + x^k))^k.

%F a(n) = Sum_{d|n} d^(d-n/d+1) * binomial(d,n/d-1).

%F If p is an odd prime, a(p) = p^p.

%t a[n_] := DivisorSum[n, #^(# - n/# + 1) * Binomial[#, n/# - 1] &]; Array[a, 20] (* _Amiram Eldar_, Aug 02 2023 *)

%o (PARI) my(N=20, x='x+O('x^N)); Vec(sum(k=1, N, (x*(k+x^k))^k))

%o (PARI) a(n) = sumdiv(n, d, d^(d-n/d+1)*binomial(d, n/d-1));

%Y Cf. A260148, A338693, A360712, A360771.

%Y Cf. A360774, A360775, A360776.

%K nonn

%O 1,2

%A _Seiichi Manyama_, Feb 20 2023