

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
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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 qseries. 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, SpringerVerlag, 1977, pp. 6978.
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 CohenOesterlé dimension formula", MathOverflow, 2014.
Jon Maiga, Computergenerated formulas for A159634, Sequence Machine.
Scanned copy of CohenOesterle.
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_{pn, p odd prime} (1 + 1/p).
a(n) = Sum_{dn, 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^2m} 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_{dm} 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*n1)^2*x^(2^n1)/(1  x^(2*n1))^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[70]) (* 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



