cp's OEIS Frontend

A106520 a(n) = A068875(n-1) - A003239(n).

%I A106520 #29 Sep 02 2024 02:35:07
%S A106520 1,0,0,0,2,4,18,48,156,472,1526,4852,16000,52940,178276,605520,
%T A106520 2079862,7201084,25138878,88358520,312576996,1112087012,3977502766,
%U A106520 14294093652,51596165872,186997738504,680272334202,2483340387644,9094756956908
%N A106520 a(n) = A068875(n-1) - A003239(n).
%C A106520 This is the multiplicity of the trivial module in a sequence of modules of dimension (2*n-2)!/n! over the symmetric groups S_n, induced from modules of dimension (2*n-2)!/(n!*(n-1)!) (Catalan) over the cyclic groups C_n.
%H A106520 G. C. Greubel, <a href="/A106520/b106520.txt">Table of n, a(n) for n = 1..1000</a>
%H A106520 F. Chapoton, <a href="http://dx.doi.org/10.2140/agt.2005.5.53">On some anticyclic operads</a>, Algebraic and Geometric Topology 5 (2005), paper no. 4, pages 53-69.
%F A106520 a(n) = (2/n) * binomial(2*n-2, n-1) - 1/(2*n) * Sum_{d divides n} phi(d) * binomial(2*n/d, n/d).
%F A106520 a(n) = 2*A000108(n-1) - (1/(2*n))*Sum_{d divides n} (n/d + 1)*A000108(n/d) * A000010(d). - _G. C. Greubel_, Aug 06 2021
%p A106520 with(numtheory);
%p A106520 a:= proc(n) (2/n)*binomial(2*n-2, n-1) - (1/(2*n))*add(phi(d)*binomial(2*n/d, n/d), d = divisors(n)) end:
%p A106520 seq(a(n), n = 1..40);
%t A106520 a[n_]:= 2/n*Binomial[2*n-2, n-1] - 1/(2*n)*DivisorSum[n, EulerPhi[#]* Binomial[2*n/#, n/#]&]; Table[a[n], {n, 40}] (* _Jean-François Alcover_, Feb 20 2017 *)
%o A106520 (Magma)
%o A106520 A106520:= func< n | 2*Catalan(n-1) - (1/(2*n))*(&+[Round(Gamma(2*n/d +1)/Gamma(n/d +1)^2)*EulerPhi(d): d in Divisors(n)]) >;
%o A106520 [A106520(n): n in [1..40]]; // _G. C. Greubel_, Aug 06 2021
%o A106520 (Sage)
%o A106520 def a(n): return 2*catalan_number(n-1) - (1/(2*n))*sum(euler_phi(n/d)*binomial(2*d, d) for d in divisors(n))
%o A106520 [a(n) for n in (1..40)] # _G. C. Greubel_, Aug 06 2021
%o A106520 (PARI) a(n) = (2/n) * binomial(2*n-2, n-1) - 1/(2*n) * sumdiv(n, d, eulerphi(d) * binomial(2*n/d, n/d)); \\ _Michel Marcus_, Aug 08 2021
%Y A106520 Cf. A000010, A000108, A001761.
%K A106520 nonn
%O A106520 1,5
%A A106520 _F. Chapoton_, May 30 2005
%E A106520 Terms a(1) to a(4) prepended by _G. C. Greubel_, Aug 06 2021