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%I #71 Nov 17 2022 05:14:26
%S 1,2,4,4,6,8,8,8,12,12,12,16,14,16,24,16,18,24,20,24,32,24,24,32,30,
%T 28,36,32,30,48,32,32,48,36,48,48,38,40,56,48,42,64,44,48,72,48,48,64,
%U 56,60,72,56,54,72,72,64,80,60,60,96,62,64,96,64,84,96,68,72,96,96
%N Coefficient for dimensions of spaces of modular & cusp forms of weight k/2, level 4*n and trivial character, where k>=5 is odd.
%C Denote dim{M_k(Gamma_0(N))} by m(k,N) and dim{S_k(Gamma_0(N))} by s(k,N).
%C We have
%C m(7/2,N)+s(5/2,N) = m(5/2,N)+s(7/2,N) =
%C (m(11/2,N)+s(9/2,N))/2 = (m(9/2,N)+s(11/2,N))/2 =
%C (m(15/2,N)+s(13/2,N))/3 = (m(13/2,N)+s(15/2,N))/3 = ...
%C (m((4j+3)/2,N)+s((4j+1)/2,N))/j = (m((4j+1)/2,N)+s((4j+3)/2,N))/j = ...
%C where N is any positive multiple of 4 and j>=1.
%C Multiplicative because A001615 is multiplicative and a(1) = A001615(2)/3 = 1. - _Andrew Howroyd_, Aug 08 2018
%D Ken Ono, The Web of Modularity: Arithmetic of Coefficients of Modular Forms and q-series. American Mathematical Society, 2004, (p. 16, theorem 1.56).
%H Peter Luschny, <a href="/A159634/b159634.txt">Table of n, a(n) for n = 1..1000</a>
%H H. Cohen and J. Oesterle, <a href="http://dx.doi.org/10.1007/BFb0065297">Dimensions des espaces de formes modulaires</a>, Modular Functions of One Variable. VI, Proc. 1976 Bonn conf., Lect. Notes in Math. 627, Springer-Verlag, 1977, pp. 69-78; <a href="https://web.archive.org/web/20060903130427/http://modular.fas.harvard.edu:9000/classes/257/references/cohen-oesterle.pdf">Scanned copy</a>.
%H S. R. Finch, <a href="/A001616/a001616.pdf">Primitive Cusp Forms</a>, April 27, 2009. [Cached copy, with permission of the author]
%H Peter Humphries, <a href="http://mathoverflow.net/questions/161948/">Answer to: "A conjecture related to the Cohen-Oesterlé dimension formula"</a>, MathOverflow, 2014.
%H Jon Maiga, <a href="http://sequencedb.net/s/A159634">Computer-generated formulas for A159634</a>, Sequence Machine.
%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Cusp_form">Cusp Form</a>.
%F a(n) = A159636(n) + A159630(n). - _Enrique Pérez Herrero_, Apr 15 2014
%F a(n) = A001615(2*n)/3. - _Enrique Pérez Herrero_, Jan 31 2014
%F From _Peter Bala_, Mar 19 2019: (Start)
%F a(n)= n*Product_{p|n, p odd prime} (1 + 1/p).
%F a(n) = Sum_{d|n, d odd} mu(d)^2*n/d, where mu(n) = A008683(n) is the Möbius function.
%F If n = m*2^k , where 2^k is the largest power of 2 dividing n, then
%F a(n) = (2^k)*a(m) = 2^k * Sum_{d^2|m} mu(d)*sigma(m/d^2), where sigma(n) = A000203(n) is the sum of the divisors of n, and also
%F a(n) = 2^k * Sum_{d|m} 2^omega(d)*phi(m/d), where omega(n) = A001221(n) is the number of different primes dividing n and phi(n) = A000010 is the Euler totient function.
%F O.g.f.: Sum_{n >= 1} mu(2*n-1)^2*x^(2^n-1)/(1 - x^(2*n-1))^2. (End)
%F a(n) = A000082(n)/A080512(n). [obvious by prime products, discovered by Sequence Machine]. - _R. J. Mathar_, Jun 24 2021
%F From _Amiram Eldar_, Nov 17 2022: (Start)
%F Multiplicative with a(2^e) = 2^e, and a(p^e) = (p+1)*p^(e-1) for p > 2.
%F Sum_{k=1..n} a(k) ~ c * n^2, where c = 6/Pi^2 = 0.607927... (A059956). (End)
%t (* per _Enrique Pérez Herrero_'s conjecture proved by P. Humphries, see link *)
%t dedekindPsi[n_Integer]:=n Apply[Times,1+1/Map[First,FactorInteger[n]]];
%t 1/3 dedekindPsi /@ (2 Range[70]) (* _Wouter Meeussen_, Apr 06 2014 *)
%o (Magma) [[4*n,(Dimension(HalfIntegralWeightForms(4*n,7/2))+ Dimension(CuspidalSubspace(HalfIntegralWeightForms(4*n,5/2))))/2] : n in [1..70]]; [[4*n,(Dimension(HalfIntegralWeightForms(4*n,5/2))+ Dimension(CuspidalSubspace(HalfIntegralWeightForms(4*n,7/2))))/2] : n in [1..70]];
%o (PARI) a(n) = 2*n*sumdiv( 2*n, d, moebius(d)^2 / d)/3; \\ _Andrew Howroyd_, Aug 08 2018
%Y Cf. A159635, A159636. - _Steven Finch_, Apr 22 2009
%Y Cf. A000010, A000082, A001221, A001615, A008683, A059956, A080512, A159630.
%K nonn,look,mult
%O 1,2
%A _Steven Finch_, Apr 17 2009
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