login
a(n) = n * Sum_{d|n} binomial(d+n-1,n)/d.
7

%I #23 Jun 14 2023 10:37:05

%S 1,5,13,49,131,545,1723,6809,24484,94445,352727,1366273,5200313,

%T 20135939,77571083,301034537,1166803127,4540794476,17672631919,

%U 68943346009,269129827042,1052178506615,4116715363823,16124644677569,63205303337656,247964681424725,973469783435197

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

%H Seiichi Manyama, <a href="/A343549/b343549.txt">Table of n, a(n) for n = 1..1000</a>

%F a(n) = [x^n] Sum_{k>=1} k * x^k/(1 - x^k)^(n+1).

%F a(n) = [x^n] Sum_{k>=1} binomial(k+n-1,n) * x^k/(1 - x^k)^2.

%F From _Seiichi Manyama_, Jun 14 2023: (Start)

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

%F a(n) = [x^n] Sum_{k>=1} (1/(1 - x^k)^n - 1). (End)

%t a[n_] := n * DivisorSum[n, Binomial[# + n - 1, n]/# &]; Array[a, 30] (* _Amiram Eldar_, Apr 25 2021 *)

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

%Y Cf. A000203, A038040, A309731, A343544, A343545, A343546, A343547, A343548.

%K nonn

%O 1,2

%A _Seiichi Manyama_, Apr 19 2021