A345230 - OEIS

%I #35 Sep 13 2024 11:59:01

%S 0,1,4,13,44,140,512,1782,6652,24682,93599,354341,1359470,5210328,

%T 20098886,77621774,300797854,1167164438,4539201401,17674941735,

%U 68933414989,269143872226,1052114789548,4116808923486,16124224585644,63205911146740,247961982954952

%N a(n) = Sum_{1 <= x_1 <= x_2 <= ... <= x_n <= n} gcd(x_1, x_2, ..., x_n).

%H Seiichi Manyama, <a href="/A345230/b345230.txt">Table of n, a(n) for n = 0..1000</a>

%F a(n) = Sum_{k=1..n} Sum_{d|k} phi(k/d) * binomial(d+n-2, n-1).

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

%F a(n) ~ 2^(2*n-1) / sqrt(Pi*n). - _Vaclav Kotesovec_, Jun 11 2021

%F a(n) = Sum_{k=1..n} phi(k) * binomial(floor(n/k)+n-1,n). - _Seiichi Manyama_, Sep 13 2024

%p a:= n-> coeff(series((1/(1-x))* add(numtheory[phi](k)

%p          *x^k/(1-x^k)^n, k=1..n), x, n+1), x, n):

%p seq(a(n), n=0..26);  # _Alois P. Heinz_, Jun 11 2021

%t a[n_] := Sum[DivisorSum[k, EulerPhi[k/#] * Binomial[n + # - 2, n - 1] &], {k, 1, n}]; Array[a, 30, 0] (* _Amiram Eldar_, Jun 11 2021 *)

%o (PARI) a(n) = sum(k=1, n, sumdiv(k, d, eulerphi(k/d)*binomial(d+n-2, n-1)));

%o (PARI) a(n) = sum(k=1, n, eulerphi(k)*binomial(n\k+n-1, n)); \\ _Seiichi Manyama_, Sep 13 2024

%Y Main diagonal of A345229.

%Y Cf. A343517, A343553, A344525.

%K nonn

%O 0,3

%A _Seiichi Manyama_, Jun 11 2021