Displaying 1-10 of 39 results found.
Coefficients in q-expansion of E_10^2, where E_10 is the Eisenstein series A013974.
+20
9
1, -528, -201168, 61114944, 20946935856, 1443146395680, 46053422547264, 861726789128832, 10894843149545520, 102119072037503664, 755968133350219680, 4623420033182073024, 24151660069581371712, 110516194189880866464, 451789196756619249792
MATHEMATICA
terms = 15;
E10[x_] = 1 - 264*Sum[k^9*x^k/(1 - x^k), {k, 1, terms}];
Exponents a(1), a(2), ... such that E_10, 1 - 264*q - 135432*q^2 + ... ( A013974) is equal to (1-q)^a(1) (1-q^2)^a(2) (1-q^3)^a(3) ... .
+20
9
264, 170148, 47083784, 21265517460, 8675419078920, 3954919534878884, 1798749087973466376, 846151096977050604564, 402076970410851910136072, 193920175271783317402925220, 94372564731126150526919627016, 46330721199213296384252696382356
COMMENTS
This sequence is related to the identity: E_4*E_6 = E_10.
Coefficients in expansion of -q*E'_10/E_10 where E_10 is the Eisenstein Series ( A013974).
+20
8
264, 340560, 141251616, 85062410400, 43377095394864, 23729517350865216, 12591243615814264896, 6769208775901467246912, 3618692733697667332476264, 1939201752717876551124987360, 1038098212042387655796115897440
FORMULA
G.f.: 1/3 * E_6/E_4 + 1/2 * E_8/E_6 - 5/6 * E_2.
MATHEMATICA
nmax = 20; Rest[CoefficientList[Series[264*x*Sum[k*DivisorSigma[9, k]*x^(k-1), {k, 1, nmax}]/(1 - 264*Sum[DivisorSigma[9, k]*x^k, {k, 1, nmax}]), {x, 0, nmax}], x]] (* Vaclav Kotesovec, Jul 09 2017 *)
G.f.: 4th root of Eisenstein series E_10 (cf. A013974).
+20
5
1, -66, -40392, -9009264, -3725341158, -1400292801072, -604993149612720, -262280205541007808, -118717180239835505592, -54520207050101542651506, -25525844887805197307977968, -12095360676632550886664063760, -5797006133905562955666277287792, -2803076705590018145443840156918512
FORMULA
a(n) ~ c * exp(2*Pi*n) / n^(5/4), where c = -3^(3/4) * Pi^(3/2) / (2^(15/4) * Gamma(3/4)^7) = -0.227361380713650977567497769428903183591275821407342369621... - Vaclav Kotesovec, Jul 02 2017, updated Mar 05 2018 G.f.: Sum_{k>=0} A004984(k) * (33*f(q))^k where f(q) is Sum_{k>=1} sigma_9(k)*q^k. - Seiichi Manyama, Jun 16 2018
MATHEMATICA
nmax = 20; s = 10; CoefficientList[Series[(1 - 2*s/BernoulliB[s] * Sum[DivisorSigma[s - 1, k]*x^k, {k, 1, nmax}])^(1/4), {x, 0, nmax}], x] (* Vaclav Kotesovec, Jul 02 2017 *)
Coefficients in expansion of -q*E'_10 where E_10 is the Eisenstein Series ( A013974).
+20
3
264, 270864, 15589728, 277365792, 2578126320, 15995060928, 74573467584, 284022573120, 920557851048, 2645157604320, 6847480097568, 16379004749184, 36394641851568, 76512377741184, 152243515448640, 290839114879104, 532222389723024, 944492355175248
Coefficients of the series whose 24th power equals E_2(x)^5/E_10(x), where E_2(x) and E_10(x) are the Eisenstein series A006352 and A013974.
+20
1
1, 6, 7038, 2002644, 922569342, 380737463400, 175255606306116, 80315525064955440, 38028486993289854966, 18171889608389845598586, 8807723964899085718419480, 4305311468773791666900669828, 2122088430918938935321961736084
COMMENTS
It is easy to see that E_2(x)^5/E_10(x) == 1 - 24*Sum_{k >= 1} (5*k - 11*k^9)*x^k/(1 - x^k) (mod 144), and also that the integer 5*k - 11*k^9 is always divisible by 6. Hence, E_2(x)^5/E_10(x) == 1 (mod 144). It follows from Heninger et al., p. 3, Corollary 2, that the series expansion of (E_2(x)^5/E_10(x))^(1/24) = 1 + 6*x + 7038*x^2 + 2002644*x^3 + 922569342*x^4 + ... has integer coefficients.
MAPLE
E(2, x) := 1 - 24*add(k*x^k/(1-x^k), k = 1..20):
E(10, x) := 1 - 264*add(k^9*x^k/(1-x^k), k = 1..20):
with(gfun): series((E(2, x)^5/E(10, x))^(1/24), x, 20):
seriestolist(%);
Expansion of Eisenstein series E_4(q) (alternate convention E_2(q)); theta series of E_8 lattice.
(Formerly M5416)
+10
186
1, 240, 2160, 6720, 17520, 30240, 60480, 82560, 140400, 181680, 272160, 319680, 490560, 527520, 743040, 846720, 1123440, 1179360, 1635120, 1646400, 2207520, 2311680, 2877120, 2920320, 3931200, 3780240, 4747680, 4905600, 6026880
COMMENTS
E_8 is also the Barnes-Wall lattice in 8 dimensions.
The E_8 lattice is integral, unimodular, and even. The 240 shortest nonzero vectors in the lattice have norm squared 2. Of these vectors, 128 are all half-integer, and 112 are all integer. - Michael Somos, Jun 10 2019
REFERENCES
J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 123.
W. Ebeling, Lattices and Codes, Vieweg; 2nd ed., 2002, see p. 53.
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 p. 111.
S. Ramanujan, On Certain Arithmetical Functions, Messenger Math., 45 (1916), 11-15 (Eq. (25)). Collected Papers of Srinivasa Ramanujan, Chap. 16, Ed. G. H. Hardy et al., Chelsea, NY, 1962.
S. Ramanujan, On Certain Arithmetical Functions, Messenger Math., 45 (1916), 11-15 (Eq. (25)). Ramanujan's Papers, p. 196, Ed. B. J. Venkatachala et al., Prism Books, Bangalore 2000.
Jean-Pierre Serre, "A Course in Arithmetic", Springer, 1978
Joseph H. Silverman, "Advanced Topics in the Arithmetic of Elliptic Curves", Springer, 1994
N. J. A. Sloane, Seven Staggering Sequences, in Homage to a Pied Puzzler, E. Pegg Jr., A. H. Schoen and T. Rodgers (editors), A. K. Peters, Wellesley, MA, 2009, pp. 93-110.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
LINKS
H. S. M. Coxeter, Integral Cayley numbers, Duke Math. J. 13 (1946), 561-578; reprinted in "Twelve Geometric Essays", pp. 20-39.
FORMULA
Can also be expressed as E4(q) = 1 + 240*Sum_{i >= 1} i^3 q^i/(1 - q^i) - Gene Ward Smith, Aug 22 2006 Theta series of E_8 lattice = 1 + 240 * Sum_{m >= 1} sigma_3(m) * q^(2*m), where sigma_3(m) is the sum of the cubes of the divisors of m ( A001158). Expansion of (phi(-q)^8 - (2 * phi(-q) * phi(q))^4 + 16 * phi(q)^8) in powers of q where phi() is a Ramanujan theta function.
Expansion of (eta(q)^24 + 256 * eta(q^2)^24) / (eta(q) * eta(q^2))^8 in powers of q. - Michael Somos, Dec 30 2008 G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^4)) where f(u, v, w) = u^2 + 33*v^2 + 256*w^2 - 18*u*v + 16*u*w - 288*v*w . - Michael Somos, Jan 05 2006 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 + 16*u2^2 + 81*u3^2 + 1296*u6^2 - 14*u1*u2 - 18*u1*u3 + 30*u1*u6 + 30*u2*u3 - 288*u2*u6 - 1134*u3*u6 . - Michael Somos, Apr 15 2007 G.f. A(x) satisfies 0 = f(A(x), A(x^3), A(x^9)) where f(u, v, w) = u^3*v + 9*w*u^3 - 84*u^2*v^2 + 246*u*v^3 - 253*v^4 - 675*w*u^2*v + 729*w^2*u^2 - 4590*w*u*v^2 + 19926*w*v^3 - 54675*w^2*u*v + 59049*w^3*u + 531441*w^3*v - 551124*w^2*v^2 . - Michael Somos, Apr 15 2007 G.f. is a period 1 Fourier series which satisfies f(-1 / t) = (t/i)^4 * f(t) where q = exp(2 Pi i t). - Michael Somos, Dec 30 2008 Expansion of Ramanujan's function Q(q^2) = 12 (omega/Pi)^4 g2 (Weierstrass invariant) in powers of q^2.
Expansion of a(q) * (a(q)^3 + 8*c(q)^3) in powers of q where a(), c() are cubic AGM theta functions. - Michael Somos, Jan 14 2015
EXAMPLE
G.f. = 1 + 240*x + 2160*x^2 + 6720*x^3 + 17520*x^4 + 30240*x^5 + 60480*x^6 + ...
G.f. = 1 + 240*q^2 + 2160*q^4 + 6720*q^6 + 17520*q^8 + 30240*q^10 + 60480*q^12 + ...
MAPLE
with(numtheory); 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(4);
MATHEMATICA
a[ n_] := If[ n < 1, Boole[n == 0], 240 DivisorSigma[ 3, n]]; (* Michael Somos, Jul 11 2011 *) a[ n_] := SeriesCoefficient[ With[ {t2 = EllipticTheta[ 2, 0, q]^4, t3 = EllipticTheta[ 3, 0, q]^4}, t2^2 + 14 t2 t3 + t3^2], {q, 0, n}]; (* Michael Somos, Jun 04 2014 *) a[ n_] := SeriesCoefficient[ With[ {t2 = EllipticTheta[ 2, 0, q]^4, t3 = EllipticTheta[ 3, 0, q]^4}, t2^2 - t2 t3 + t3^2], {q, 0, 2 n}]; (* Michael Somos, Jul 31 2016 *)
PROG
(PARI) {a(n) = if( n<1, n==0, 240 * sigma(n, 3))};
(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x + A)^24 + 256 * x * eta(x^2 + A)^24) / (eta(x + A) * eta(x^2 + A))^8, n))}; /* Michael Somos, Dec 30 2008 */ (PARI) q='q+O('q^50); Vec((eta(q)^24+256*q*eta(q^2)^24)/(eta(q)*eta(q^2))^8) \\ Altug Alkan, Sep 30 2018 (Sage) ModularForms(Gamma1(1), 4, prec=30).0 ; # Michael Somos, Jun 04 2013 (Magma) Basis( ModularForms( Gamma1(1), 4), 29) [1]; /* Michael Somos, May 11 2015 */ (Magma) L := Lattice("E", 8); A<q> := ThetaSeries(L, 57); A; /* Michael Somos, Jun 10 2019 */ (Python)
from sympy import divisor_sigma
def a(n): return 1 if n == 0 else 240 * divisor_sigma(n, 3)
Expansion of Eisenstein series E_6(q) (alternate convention E_3(q)).
+10
149
1, -504, -16632, -122976, -532728, -1575504, -4058208, -8471232, -17047800, -29883672, -51991632, -81170208, -129985632, -187132176, -279550656, -384422976, -545530104, -715608432, -986161176, -1247954400, -1665307728, -2066980608, -2678616864, -3243917376, -4159663200
REFERENCES
W. Ebeling, Lattices and Codes, Vieweg; 2nd ed., 2002, see p. 53.
R. C. Gunning, Lectures on Modular Forms. Princeton Univ. Press, Princeton, NJ, 1962, p. 53.
M. Kaneko and D. Zagier, Supersingular j-invariants, hypergeometric series and Atkin's orthogonal polynomials, pp. 97-126 of D. A. Buell and J. T. Teitelbaum, eds., Computational Perspectives on Number Theory, Amer. Math. Soc., 1998.
N. Koblitz, Introduction to Elliptic Curves and Modular Forms, Springer-Verlag, 1984, see p. 111.
Jean-Pierre Serre, "A Course in Arithmetic", Springer, 1978
Joseph H. Silverman, "Advanced Topics in the Arithmetic of Elliptic Curves", Springer, 1994
FORMULA
E6(q) = 1 - 504*Sum_{i>=1} sigma_5(i)q^i where sigma_5(n) is A001160, the sum of fifth powers of the divisors of n. It can also be expressed as E6(q) = 1 - 504*Sum_{i>=1} i^5*q^i/(1-q^i). - Gene Ward Smith, Aug 22 2006 G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^4)) where f(u, v, w) = u^2*v - 8*u^2*w - 66*u*v^2 + 592*u*v*w - 512*u*w^2 + 121*v^3 - 4224*v^2*w + 4096*v*w^2. - Michael Somos, Apr 10 2005 Expansion of Ramanujan's function R(q) = 216*g3 (Weierstrass invariant).
Expansion of (eta(q)^8 + 32 * eta(q^4)^8) * (eta(q)^16 - 512 * eta(q)^8 * eta(q^4)^8 - 8192 * eta(q^4)^16) / eta(q^2)^12 in powers of q. - Michael Somos, Dec 30 2008 G.f. is a period 1 Fourier series which satisfies f(-1 / t) = - (t/i)^6 * f(t) where q = exp(2 Pi i t). - Michael Somos, Dec 30 2008 E6(q) = eta(q)^24 / eta(q^2)^12 - 480 * eta(q^2)^12 - 16896 * eta(q^2)^12 * eta(q^4)^8 / eta(q)^8 + 8192 * eta(q^4)^24 / eta(q^2)^12. - Seiichi Manyama, May 08 2017
EXAMPLE
G.f. = 1 - 504*q - 16632*q^2 - 122976*q^3 - 532728*q^4 - 1575504*q^5 + ...
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(6);
# alternative
if n = 0 then
1;
else
-504*numtheory[sigma][5](n) ;
end if;
end proc:
MATHEMATICA
a[ n_] := If[ n < 1, Boole[n == 0], -504 DivisorSigma[ 5, n]]; (* Michael Somos, Apr 21 2013 *) a[ n_] := SeriesCoefficient[ With[ {t2 = EllipticTheta[ 2, 0, q]^4, t3 = EllipticTheta[ 3, 0, q]^4}, t2^3 - 33 (t2 + t3) t2 t3 + t3^3], {q, 0, n}]; (* Michael Somos, Apr 21 2013 *) a[ n_] := SeriesCoefficient[ With[ {t3 = EllipticTheta[ 3, 0, q]^4, t4 = EllipticTheta[ 4, 0, q]^4}, (t3^3 - 3 (t3 - t4)^2 (t3 + t4) + t4^3) / 2], {q, 0, 2 n}]; (* Michael Somos, Jun 04 2014 *) a[ n_] := SeriesCoefficient[ With[ {e1 = QPochhammer[ q]^8, e4 = 32 q QPochhammer[ q^4]^8}, (e1 + e4) (e1^2 - 16 e1 e4 - 8 e4^2) / QPochhammer[ q^2]^12], {q, 0, n}]; (* Michael Somos, Apr 01 2015 *) a[ n_] := SeriesCoefficient[ With[ {t2 = EllipticTheta[ 2, 0, q]^4, t3 = EllipticTheta[ 3, 0, q]^4}, t2^3 - 3/2 (t2 + t3) t2 t3 + t3^3], {q, 0, 2 n}]; (* Michael Somos, Jul 31 2016 *) terms = 25; E6[x_] = 1-(12/BernoulliB[6])*Sum[k^5*x^k/(1-x^k), {k, terms}]; CoefficientList[E6[x] + O[x]^terms, x] (* Jean-François Alcover, Feb 28 2018 *)
PROG
(PARI) {a(n) = if( n<1, n==0, -504 * sigma( n, 5))};
(PARI) {a(n) = my(A, A1, A4); if( n<0, 0, A = x * O(x^n); A1 = eta(x + A)^8; A4 = 32 * x * eta(x^4 + A)^8; polcoeff( (A1 + A4) * (A1^2 - 16 * A1 * A4 - 8 * A4^2) / eta(x^2 + A)^12, n))}; /* Michael Somos, Dec 30 2008 */ (Sage) ModularForms( Gamma1(1), 6, prec=25).0; # Michael Somos, Jun 04 2013 (Magma) Basis( ModularForms( Gamma1(1), 6), 25); /* Michael Somos, Apr 01 2015 */
Coefficients in expansion of Eisenstein series E_2 (also called E_1 or G_2).
(Formerly M5145)
+10
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
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).
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)
a(0) = 1, a(n) = 480*sigma_7(n).
+10
44
1, 480, 61920, 1050240, 7926240, 37500480, 135480960, 395301120, 1014559200, 2296875360, 4837561920, 9353842560, 17342613120, 30119288640, 50993844480, 82051050240, 129863578080, 196962563520
COMMENTS
Eisenstein series E_8(q) (alternate convention E_4(q)); theta series of direct sum of 2 copies of E_8 lattice.
REFERENCES
J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 123.
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 p. 111.
S. Ramanujan, On Certain Arithmetical Functions, Messenger Math., 45 (1916), 11-15 (Eq. (25)). Collected Papers of Srinivasa Ramanujan, Chap. 16, Ed. G. H. Hardy et al., Chelsea, NY, 1962.
S. Ramanujan, On Certain Arithmetical Functions, Messenger Math., 45 (1916), 11-15 (Eq. (25)). Ramanujan's Papers, p. 196, Ed. B. J. Venkatachala et al., Prism Books, Bangalore 2000.
FORMULA
Equivalently, g.f. = (theta2^16+theta3^16+theta4^16)/2.
G.f. Sum{k>=0} a(k)q^(2k) = (theta2^16+theta3^16+theta4^16)/2.
Expansion of ((eta(q)^24 + 256 * eta(q^2)^24) / (eta(q) * eta(q^2))^8)^2 in powers of q. - Michael Somos, Dec 30 2008 G.f. is a period 1 Fourier series which satisfies f(-1 / t) = (t/i)^8 * f(t) where q = exp(2 Pi i t). - Michael Somos, Dec 30 2008
EXAMPLE
G.f. = 1 + 480*q + 61920*q^2 + 1050240*q^3 + 7926240*q^4 + 37500480*q^5 + ...
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(8);
MATHEMATICA
a[ n_] := If[ n < 1, Boole[n == 0], 480 DivisorSigma[ 7, n]]; (* Michael Somos, Jun 04 2013 *) nmax = 60; CoefficientList[Series[(Product[(1-x^k)^8 / (1+x^k)^8, {k, 1, nmax}] + 256 * x * Product[(1+x^k)^16 *(1-x^k)^8, {k, 1, nmax}])^2, {x, 0, nmax}], x] (* Vaclav Kotesovec, Jul 02 2017 *)
PROG
(PARI) {a(n) = if( n<1, n==0, 480 * sigma(n, 7))};
(PARI) {a(n) = local(A, e1, e2, e4); if( n<0, 0, n*=2; A = x * O(x^n); e1 = eta(x + A)^16; e2 = eta(x^2 + A)^16; e4 = eta(x^4 + A)^16; polcoeff( (e1*e2^3 + 256*x^2 * e4*(e2^3 + e1^2*e4)) / (e1*e2*e4), n))}; /* Michael Somos, Jun 29 2005 */ (PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( ((eta(x + A)^24 + 256 * x * eta(x^2 + A)^24) / (eta(x + A) * eta(x^2 + A))^8)^2, n))}; /* Michael Somos, Dec 30 2008 */ (Sage) ModularForms( Gamma1(1), 8, prec=33).0; # Michael Somos, Jun 04 2013 (Magma) Basis( ModularForms( Gamma1(1), 8), 33) [1]; /* Michael Somos, May 27 2014 */
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