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A063118 Dimension of the space of weight 2n cusp forms for Gamma_0(50).

%I A063118 #50 Sep 26 2024 15:09:04
%S A063118 2,17,31,47,61,77,91,107,121,137,151,167,181,197,211,227,241,257,271,
%T A063118 287,301,317,331,347,361,377,391,407,421,437,451,467,481,497,511,527,
%U A063118 541,557,571,587,601,617,631,647,661,677,691,707,721,737,751,767,781
%N A063118 Dimension of the space of weight 2n cusp forms for Gamma_0(50).
%C A063118 Appears to agree with the first 11-section of A186042 except for the first term of both sequences (verified up to a(10000)). - _Klaus Brockhaus_, Mar 10 2011
%H A063118 Klaus Brockhaus, <a href="/A063118/b063118.txt">Table of n, a(n) for n = 1..10000</a>
%H A063118 William A. Stein, <a href="http://wstein.org/Tables/dimskg0n.gp">Dimensions of the spaces S_k(Gamma_0(N))</a>.
%H A063118 William A. Stein, <a href="http://wstein.org/Tables/">The modular forms database</a>
%F A063118 From _Klaus Brockhaus_, Mar 10 2011: (Start)
%F A063118 G.f. (conjectured): x*(x^3 + 12*x^2 + 15*x + 2) / ((x - 1)^2*(x + 1)).
%F A063118 Recurrences (conjectured):
%F A063118 a(n) = a(n-1) + a(n-2) - a(n-3) for n > 4;
%F A063118 a(n) = a(n-2) + 30 for n > 3. (End)
%F A063118 Closed formula (conjectured): a(n) = (30*n+(-1)^n-27)/2 for n > 1. - _Bruno Berselli_, Mar 10 2011
%F A063118 Recurrence (conjectured): a(n) = 2*a(n-1) -a(n-2) +2*(-1)^n, n > 3. - _Vincenzo Librandi_, Mar 24 2011
%F A063118 Conjecture: a(n) = A007775(4*n - 3), n > 1. - _Bill McEachen_, May 15 2022
%e A063118 G.f. = 2*x + 17*x^2 + 31*x^3 + 47*x^4 + 61*x^5 + 77*x^6 + 91*x^7 + 107*x^8 + 121*x^9 + ...
%o A063118 (Magma) [ Dimension(CuspForms(Gamma0(50), 2*n)): n in [1..55] ]; // _Klaus Brockhaus_, Mar 10 2011
%o A063118 (Sage) def a(n) : return( len( CuspForms( Gamma0( 50), 2*n, prec=1) . basis())); # _Michael Somos_, May 29 2013
%Y A063118 Cf. A186042, A007775.
%K A063118 nonn
%O A063118 1,1
%A A063118 _N. J. A. Sloane_, Jul 08 2001