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 A159634 Coefficient for dimensions of spaces of modular & cusp forms of weight k/2, level 4*n and trivial character, where k>=5 is odd. 5
 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, 28, 36, 32, 30, 48, 32, 32, 48, 36, 48, 48, 38, 40, 56, 48, 42, 64, 44, 48, 72, 48, 48, 64, 56, 60, 72, 56, 54, 72, 72, 64, 80, 60, 60, 96, 62, 64, 96, 64, 84, 96, 68, 72, 96, 96 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS Denote dim{M_k(Gamma_0(N))} by m(k,N) and dim{S_k(Gamma_0(N))} by s(k,N). We have m(7/2,N)+s(5/2,N) = m(5/2,N)+s(7/2,N) = (m(11/2,N)+s(9/2,N))/2 = (m(9/2,N)+s(11/2,N))/2 = (m(15/2,N)+s(13/2,N))/3 = (m(13/2,N)+s(15/2,N))/3 = ... (m((4j+3)/2,N)+s((4j+1)/2,N))/j = (m((4j+1)/2,N)+s((4j+3)/2,N))/j = ... where N is any positive multiple of 4 and j>=1. Multiplicative because A001615 is multiplicative and a(1) = A001615(2)/3 = 1. - Andrew Howroyd, Aug 08 2018 REFERENCES K. Ono, The Web of Modularity: Arithmetic of Coefficients of Modular Forms and q-series. American Mathematical Society, 2004, (p. 16, theorem 1.56). LINKS Peter Luschny, Table of n, a(n) for n = 1..1000 H. Cohen and J. Oesterle, Dimensions des espaces de formes modulaires, Modular Functions of One Variable. VI, Proc. 1976 Bonn conf., Lect. Notes in Math. 627, Springer-Verlag, 1977, pp. 69-78. S. R. Finch, Primitive Cusp Forms, April 27, 2009. [Cached copy, with permission of the author] Peter Humphries, Answer to: "A conjecture related to the Cohen-Oesterlé dimension formula", MathOverflow, 2014. Jon Maiga, Computer-generated formulas for A159634, Sequence Machine. Wikipedia, Cusp Form FORMULA a(n) = A159636(n) + A159630(n). - Enrique Pérez Herrero, Apr 15 2014 a(n) = A001615(2*n)/3. - Enrique Pérez Herrero, Jan 31 2014 From Peter Bala, Mar 19 2019: (Start) a(n)= n*Product_{p|n, p odd prime} (1 + 1/p). a(n) = Sum_{d|n, d odd} mu(d)^2*n/d, where mu(n) = A008683(n) is the Möbius function. If n = m*2^k , where 2^k is the largest power of 2 dividing n, then 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 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. O.g.f.: Sum_{n >= 1} mu(2*n-1)^2*x^(2^n-1)/(1 - x^(2*n-1))^2. (End) a(n) = A000082(n)/A080512(n). [obvious by prime products, discovered by Sequence Machine]. - R. J. Mathar, Jun 24 2021 MATHEMATICA (* per Enrique Pérez Herrero's conjecture proved by P. Humphries, see link *) dedekindPsi[n_Integer]:=n Apply[Times, 1+1/Map[First, FactorInteger[n]]]; 1/3 dedekindPsi /@ (2 Range) (* Wouter Meeussen, Apr 06 2014 *) PROG (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]]; (PARI) a(n) = 2*n*sumdiv( 2*n, d, moebius(d)^2 / d)/3; \\ Andrew Howroyd, Aug 08 2018 CROSSREFS Cf. A159635, A159636. - Steven Finch, Apr 22 2009 Sequence in context: A288529 A288772 A053196 * A186690 A002131 A230641 Adjacent sequences:  A159631 A159632 A159633 * A159635 A159636 A159637 KEYWORD nonn,look,mult AUTHOR Steven Finch, Apr 17 2009 STATUS approved

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Last modified September 26 01:31 EDT 2022. Contains 356986 sequences. (Running on oeis4.)