|
|
A006352
|
|
Coefficients in expansion of Eisenstein series E_2 (also called E_1 or G_2).
(Formerly M5145)
|
|
108
|
|
|
1, -24, -72, -96, -168, -144, -288, -192, -360, -312, -432, -288, -672, -336, -576, -576, -744, -432, -936, -480, -1008, -768, -864, -576, -1440, -744, -1008, -960, -1344, -720, -1728, -768, -1512, -1152, -1296, -1152, -2184, -912, -1440, -1344, -2160, -1008, -2304, -1056, -2016, -1872, -1728
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
0,2
|
|
COMMENTS
|
The series Q(q), R(q) are modular forms, but P(q) is not. - Michael Somos, May 18 2017
|
|
REFERENCES
|
R. C. Gunning, Lectures on Modular Forms. Princeton Univ. Press, Princeton, NJ, 1962, p. 53.
N. Koblitz, Introduction to Elliptic Curves and Modular Forms, Springer-Verlag, 1984, see pp. 111 and 113.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
Zagier, Don. "Elliptic modular forms and their applications." The 1-2-3 of modular forms. Springer Berlin Heidelberg, 2008. 1-103. See p. 19, Eq. (17).
|
|
LINKS
|
|
|
FORMULA
|
a(n) = -24*sigma(n) = -24*A000203(n), for n>0.
G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^3), A(x^6)) where f(u1, u2, u3, u6) = u1^2 + 4*u2^2 + 9*u3^2 + 36*u6^2 - 8*u1*u2 + 6*u1*u3 + 24*u2*u6 - 72*u3*u6. - Michael Somos, May 29 2005
G.f.: 1 - 24*sum(k>=1, k*x^k/(1 - x^k)).
G.f.: 1 + 24 *x*deriv(eta(x))/eta(x) where eta(x) = prod(n>=1, 1-x^n); (cf. A000203). - Joerg Arndt, Sep 28 2012
G.f.: 1 - 24*x/(1-x) + 48*x^2/(Q(0) - 2*x^2 + 2*x), where Q(k)= (2*x^(k+2) - x - 1)*k - 1 - 2*x + 3*x^(k+2) - x*(k+1)*(k+3)*(1-x^(k+2))^2/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, May 16 2013
G.f.: q*Delta'/Delta where Delta is the generating function of Ramanujan's tau function (A000594). - Seiichi Manyama, Jul 15 2017
|
|
EXAMPLE
|
G.f. = 1 - 24*x - 72*x^2 - 96*x^3 - 168*x^4 - 144*x^5 - 288*x^6 + ...
|
|
MAPLE
|
E := proc(k) local n, t1; t1 := 1-(2*k/bernoulli(k))*add(sigma[k-1](n)*q^n, n=1..60); series(t1, q, 60); end; E(2);
|
|
MATHEMATICA
|
a[n_] := -24*DivisorSigma[1, n]; a[0] = 1; Table[a[n], {n, 0, 46}] (* Jean-François Alcover, Dec 12 2012 *)
a[ n_] := If[ n < 1, Boole[n == 0], -24 DivisorSigma[ 1, n]]; (* Michael Somos, Apr 08 2015 *)
|
|
PROG
|
(PARI) {a(n) = if( n<1, n==0, -24 * sigma(n))}; /* Michael Somos, Apr 09 2003 */
(Python)
from sympy import divisor_sigma
def a(n): return 1 if n == 0 else -24 * divisor_sigma(n)
|
|
CROSSREFS
|
|
|
KEYWORD
|
sign,easy,nice
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|