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A081875 a(n) = Sum_{d|n} phi(n/d)*C(2*d,d)/2.

%I A081875 #28 Nov 29 2024 16:32:40
%S A081875 1,4,12,40,130,480,1722,6480,24336,92520,352726,1352640,5200312,
%T A081875 20060040,77559060,300546720,1166803126,4537592928,17672631918,
%U A081875 68923357200,269128940724,1052049834616,4116715363822,16123803207552
%N A081875 a(n) = Sum_{d|n} phi(n/d)*C(2*d,d)/2.
%H A081875 Robert Israel, <a href="/A081875/b081875.txt">Table of n, a(n) for n = 1..1656</a>
%H A081875 V. A. Liskovets, <a href="https://doi.org/10.37236/1841">Enumerative formulas for unrooted planar maps: a pattern</a>, Electron. J. Combin., 11:1 (2004), R88.
%H A081875 J. Malenfant, <a href="http://arxiv.org/abs/1502.06012">On the Matrix-Element Expansion of a Circulant Determinant</a>, arXiv preprint arXiv:1502.06012 [math.NT], 2015.
%F A081875 Dirichlet convolution of A000010 and A088218. - _R. J. Mathar_, Mar 11 2017
%F A081875 a(n) ~ 2^(2*n-1) / sqrt(Pi*n). - _Vaclav Kotesovec_, Jun 08 2019
%F A081875 From _Richard L. Ollerton_, May 07 2021: (Start)
%F A081875 a(n) = Sum_{k=1..n} C(2*gcd(n,k),gcd(n,k))/2.
%F A081875 a(n) = Sum_{k=1..n} A088218(gcd(n,k)). (End)
%e A081875 G.f. = x + 4*x^2 + 12*x^3 + 40*x^4 + 130*x^5 + 480*x^6 + 1722*x^7 + ...
%p A081875 f:= proc(n) local d; add(numtheory:-phi(n/d)*binomial(2*d,d)/2, d = numtheory:-divisors(n)) end proc:
%p A081875 map(f, [$1..30]); # _Robert Israel_, Nov 29 2024
%t A081875 Table[Fold[ #1+EulerPhi[n/#2]*Binomial[2#2, #2]/2&, 0, Divisors[n]], {n, 1, 32}]
%t A081875 a[ n_] := If[ n < 0, 0, Sum[ Binomial[2 d, d] EulerPhi[n / d], {d, Divisors @ n}] / 2]; (* _Michael Somos_, Nov 01 2014 *)
%o A081875 (PARI) {a(n) = if( n<1, 0, sumdiv(n, d, binomial(2*d, d) * eulerphi(n/d)) / 2)}; /* _Michael Somos_, Nov 01 2014 */
%Y A081875 Cf. A000010, A045630, A088218.
%K A081875 easy,nonn
%O A081875 1,2
%A A081875 _Wouter Meeussen_, Apr 12 2003