A013973 -id:A013973 - cp's OEIS frontend

A037946 Coefficients of unique normalized cusp form Delta_22 of weight 22 for full modular group.

1, -288, -128844, -2014208, 21640950, 37107072, -768078808, 1184071680, 6140423133, -6232593600, -94724929188, 259518615552, -80621789794, 221206696704, -2788306561800, 3883087691776, 3052282930002

Offset: 1

N. J. A. Sloane

sign

			q^2 - 288*q^4 - ...

G. Harder. "A Congruence Between a Siegel and an Elliptic Modular Form." The 1-2-3 of modular forms. Springer Berlin Heidelberg, 2008. 247-262.

Seiichi Manyama, Table of n, a(n) for n = 1..1000
Fernando Q. Gouvêa, Non-ordinary primes: a story, Experimental Mathematics, Volume 6, Issue 3 (1997), 195-205.
S. C. Milne, Hankel determinants of Eisenstein series, preprint, arXiv:0009130 [math.NT], 2000.
Index entries for sequences related to modular groups

Cf. A000594 ((E_4(q)^3 - E_6(q)^2)/12^3), A004009 (E_4(q)), A013969, A013973 (E_6(q)), A290181.

terms = 17;
E4[x_] = 1 + 240*Sum[k^3*x^k/(1 - x^k), {k, 1, terms+1}];
E6[x_] = 1 - 504*Sum[k^5*x^k/(1 - x^k), {k, 1, terms+1}];
((E4[x]^3 - E6[x]^2)/12^3)*E4[x]*E6[x] + O[x]^(terms+1) // CoefficientList[#, x]& // Rest (* Jean-François Alcover, Feb 27 2018, after Seiichi Manyama *)

a(n) == A013969(n) mod 77683. - Seiichi Manyama, Feb 03 2017
G.f.: (E_4(q)^3 - E_6(q)^2)/12^3 * E_4(q) * E_6(q). - Seiichi Manyama, Jun 09 2017
G.f.: 691/(1728*250) * (E_8(q)*E_14(q) - E_10(q)*E_12(q)). - Seiichi Manyama, Jul 25 2017

A008689 Theta series of Niemeier lattice of type A_24.

Original entry on oeis.org

1, 600, 182160, 16924320, 397150800, 4632279120, 34414027200, 187479888960, 814930462800, 2975483302200, 9486481747680, 27053266687200, 70486014432960, 169930748683920, 384163827465600

Offset: 0

N. J. A. Sloane

nonn

J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 407.

G. C. Greubel, Table of n, a(n) for n = 0..1000

Cf. A004009, A013973, A023936.

terms = 15; E4[q_] := 1 + 240 Sum[DivisorSigma[3, n]*q^(2 n), {n, 1, terms}]; E6[q_] := 1 - 504 Sum[DivisorSigma[5, n]*q^(2 n), {n, 1, terms}]; s = 67/72 E4[q]^3 + 5/72 E6[q]^2 + O[q]^(3 terms); Partition[ CoefficientList[s, q], 2][[All, 1]][[1 ;; terms]] (* Jean-François Alcover, Jul 06 2017 *)

a(n) = A023936(25n). - Kok Seng Chua (chuaks(AT)ihpc.nus.edu.sg), Aug 02 2001

This series is the q-expansion of 67/72 E_4(z)^3 + 5/72 E_6(z)^2. - Daniel D. Briggs, Nov 25 2011

A008691 Theta series of Niemeier lattice of type A_17 E_7.

Original entry on oeis.org

1, 432, 186192, 16881984, 397398096, 4631467680, 34415043264, 187482701952, 814916270160, 2975502394224, 9486501222240, 27053176872384, 70486076751552, 169930845743904, 384163759953792, 820167146628480, 1668890516764752, 3249628128869472, 6096883839494544

Offset: 0

N. J. A. Sloane

nonn

Also the theta series for the Niemeier lattice of type D_10 E_7^2. - clarified by Ben Mares, Jul 17 2022

J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 407.

G. C. Greubel, Table of n, a(n) for n = 0..1000

Cf. A004009, A013973.

terms = 15; E4[q_] := 1 + 240 Sum[DivisorSigma[3, n]*q^(2 n), {n, 1, terms}]; E6[q_] := 1 - 504 Sum[DivisorSigma[5, n]*q^(2 n), {n, 1, terms}]; s = 5/6 E4[q]^3 + 1/6 E6[q]^2 + O[q]^(3 terms); Partition[ CoefficientList[s, q], 2][[All, 1]][[1 ;; terms]] (* Jean-François Alcover, Jul 06 2017 *)

This series is the q-expansion of 5/6 E_4(z)^3 + 1/6 E_6(z)^2. See A004009 and A013973. - Daniel D. Briggs, Nov 25 2011

More terms from Sean A. Irvine, Mar 22 2020

A008695 Theta series of Niemeier lattice of type A_11 D_7 E_6.

Original entry on oeis.org

1, 288, 189648, 16845696, 397610064, 4630772160, 34415914176, 187485113088, 814904105040, 2975518758816, 9486517914720, 27053099888256, 70486130167488, 169930928938176, 384163702086528

Offset: 0

N. J. A. Sloane

nonn

Also the theta series of the Niemeier lattice of type E_6^4. - clarified by Ben Mares, Sep 13 2022

J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 407.

G. C. Greubel, Table of n, a(n) for n = 0..1000

Cf. A004009, A013973.

Cf. A008688 - A008694, A008696 - A008704.

terms = 15; th = EllipticTheta; E4 = 1 + 240*Sum[k^3*(q^k/(1 - q^k)), {k, 1, terms}] + O[q]^terms; E6 = th[2, 0, q]^12 + th[3, 0, q]^12 - 33*th[2, 0, q]^4*th[3, 0, q]^4*(th[2, 0, q]^4 + th[3, 0, q]^4); CoefficientList[ (3/4)*E4^3 + (1/4)*E6^2 + O[q]^terms, q] (* Jean-François Alcover, Jul 05 2017 *)

This series is the q-expansion of (3*E_4(z)^3 + E_6(z)^2)/4. - Daniel D. Briggs, Nov 25 2011

A008696 Theta series of Niemeier lattice of type D_6^4.

Original entry on oeis.org

1, 240, 190800, 16833600, 397680720, 4630540320, 34416204480, 187485916800, 814900050000, 2975524213680, 9486523478880, 27053074226880, 70486147972800, 169930956669600, 384163682797440, 820166912933760

Offset: 0

N. J. A. Sloane

nonn

Also the theta series for the Niemeier lattice of type A_9^2 D_6. - clarified by Ben Mares, Sep 13 2022

J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 407.

G. C. Greubel, Table of n, a(n) for n = 0..1000

Cf. A004009, A013973.

Cf. A008688 - A008695, A008697 - A008704.

terms = 15; th = EllipticTheta; E4 = 1 + 240*Sum[k^3*(q^k/(1 - q^k)), {k, 1, terms}] + O[q]^terms; E6 = th[2, 0, q]^12 + th[3, 0, q]^12 - 33*th[2, 0, q]^4*th[3, 0, q]^4*(th[2, 0, q]^4 + th[3, 0, q]^4); CoefficientList[ (13/18)*E4^3 + (5/18)*E6^2 + O[q]^terms, q] (* Jean-François Alcover, Jul 05 2017 *)

This series is the q-expansion of (13*E_4(z)^3 + 5*E_6(z)^2)/18. - Daniel D. Briggs, Nov 25 2011

A008700 Theta series of Niemeier lattice of type D_4^6.

Original entry on oeis.org

1, 144, 193104, 16809408, 397822032, 4630076640, 34416785088, 187487524224, 814891939920, 2975535123408, 9486534607200, 27053022904128, 70486183583424, 169931012132448, 384163644219264, 820166796086400

Offset: 0

N. J. A. Sloane

nonn

Also the theta series of the Niemeier lattice of type A_5^4 D_4. - clarified by Ben Mares, Jul 17 2022

J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 407.

G. C. Greubel, Table of n, a(n) for n = 0..1000

Cf. A004009, A013973.

Cf. A008688 - A008699, A008701, A008702, A008703, A008704.

terms = 15; th = EllipticTheta; E4 = 1 + 240*Sum[k^3*(q^k/(1 - q^k)), {k, 1, terms}] + O[q]^terms; E6 = th[2, 0, q]^12 + th[3, 0, q]^12 - 33*th[2, 0, q]^4*th[3, 0, q]^4*(th[2, 0, q]^4 + th[3, 0, q]^4); CoefficientList[ (2/3)*E4^3 + (1/3)*E6^2 + O[q]^terms, q] (* Jean-François Alcover, Jul 05 2017 *)

This series is the q-expansion of (2*E_4(z)^3 + E_6(z)^2)/3. - Daniel D. Briggs, Nov 25 2011

A008702 Theta series of Niemeier lattice of type A_3^8.

Original entry on oeis.org

1, 96, 194256, 16797312, 397892688, 4629844800, 34417075392, 187488327936, 814887884880, 2975540578272, 9486540171360, 27052997242752, 70486201388736, 169931039863872, 384163624930176

Offset: 0

N. J. A. Sloane

nonn

J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 407.

G. C. Greubel, Table of n, a(n) for n = 0..1000

Cf. A004009, A013973.

Cf. A008688 - A008701, A008703, A008704.

terms = 15; E4[q_] := 1 + 240 Sum[DivisorSigma[3, n]*q^(2 n), {n, 1, terms}]; E6[q_] := 1 - 504 Sum[DivisorSigma[5, n]*q^(2 n), {n, 1, terms}]; s = 23/36 E4[q]^3 + 13/36 E6[q]^2 + O[q]^(3 terms); Partition[ CoefficientList[s, q], 2][[All, 1]][[1 ;; terms]] (* Jean-François Alcover, Jul 06 2017 *)

This series is the q-expansion of (23*E_4(z)^3 + 13*E_6(z)^2)/36. - Daniel D. Briggs, Nov 26 2011

A126858 Coefficients in quasimodular form F_2(q) of level 1 and weight 6.

Original entry on oeis.org

0, 0, 1, 8, 30, 80, 180, 336, 620, 960, 1590, 2200, 3416, 4368, 6440, 7920, 11160, 13056, 18333, 20520, 27860, 31360, 41052, 44528, 59760, 62400, 80990, 87120, 109872, 113680, 147960, 148800, 188976, 196416, 240210, 243040, 311910, 303696, 376580, 385840

Offset: 0

N. J. A. Sloane, Mar 15 2007

nonn

This is also (5*E_2^3 - 3*E_2*E_4 - 2*E_6)/51840, where E_2, E_4, E_6 are the Eisenstein series shown in A006352, A004009, A013973, respectively. - N. J. A. Sloane, Feb 06 2017

This is also ((q*(d/dq)E_4)/240 + q*(d/dq)(q*(d/dq)E_2)/24)/6, where E_2 and E_4 are the Eisenstein series shown in A006352 and A004009, respectively. - Seiichi Manyama, Feb 08 2017

			F_2(q) = q^2 + 8*q^3 + 30*q^4 + 80*q^5 + 180*q^6 + 336*q^7 + 620*q^8 + 960*q^9 + 1590*q^10 + 2200*q^11 + ...

Seiichi Manyama, Table of n, a(n) for n = 0..1000
Michael Andrew Henry, Hecke vector-forms: vector functions of quasiautomorphic forms over Hecke triangle groups, ResearchGate (2025). See p. 14.
B. Mazur, Perturbations, deformations and variations (and "near-misses") in geometry, physics, and number theory, Bull. Amer. Math. Soc., 41 (2004), 307-336.

Cf. A006352 (E_2), A004009 (E_4), A013973 (E_6), A145094 (q*(d/dq)E_4), A281372, A282097, A282154 (-q*(d/dq)(q*(d/dq)E_2)).

with(numtheory); M:=100;
E := proc(k) local n, t1; global M;
t1 := 1-(2*k/bernoulli(k))*add(sigma[k-1](n)*q^n, n=1..M+1);
series(t1, q, M+1); end;
e2:=E(2); e4:=E(4); e6:=E(6);
series((5*e2^3-3*e2*e4-2*e6)/51840,q,M+1);
seriestolist(%); # from N. J. A. Sloane, Feb 06 2017

terms = 40; Ei[n_] = 1 - (2 n/BernoulliB[n]) Sum[k^(n-1) x^k/(1-x^k), {k, 1, terms}]; S = 5 Ei[2]^3 - 3 Ei[2] Ei[4] - 2 Ei[6]; CoefficientList[S + O[x]^terms, x]/SeriesCoefficient[S, {x, 0, 2}] (* Jean-François Alcover, Feb 28 2018 *)

{a(n) = local(L1, L2, L3); if( n<0, 0, L1 = 1 - 24 * sum( k = 1, n, sigma(k) * x^k, x * O(x^n)); L2 = x * L1'; L3 = x * L2'; polcoeff( (L1 * L2 - L3) / 720, n))} /* Michael Somos, Jan 08 2012 */

F_2(q) = (5*E(2)^3-3*E(2)*E(4)-2*E(6))/51840 where E(k) is the normalized Eisenstein series of weight k (cf. A006352, etc.).
Expansion of (L1 * L2 - L3) / 720 where L1 = E2 (A006352), L2 = q * dL1/dq, L3 = q * dL2/dq in powers of q where E2 is an Eisenstein series. - Michael Somos, Jan 08 2012
a(n) = (A145094(n)/240 - A282154(n)/24)/6 = (A281372(n) - A282097(n))/6. - Seiichi Manyama, Feb 08 2017

A186100 Expansion of 2 * a(q^2)^2 - a(q)^2 in powers of q where a() is a cubic AGM theta function.

Original entry on oeis.org

1, -12, -12, -12, -12, -72, -12, -96, -12, -12, -72, -144, -12, -168, -96, -72, -12, -216, -12, -240, -72, -96, -144, -288, -12, -372, -168, -12, -96, -360, -72, -384, -12, -144, -216, -576, -12, -456, -240, -168, -72, -504, -96, -528, -144, -72, -288

Offset: 0

Michael Somos, Feb 12 2011

sign

Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).
Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
Ramanujan's Eisenstein series: P(q) (see A006352), Q(q) (A004009), R(q) (A013973).

			G.f. = 1 - 12*q - 12*q^2 - 12*q^3 - 12*q^4 - 72*q^5 - 12*q^6 - 96*q^7 + ...

G. C. Greubel, Table of n, a(n) for n = 0..1000
J. M. Borwein and P. B. Borwein, A cubic counterpart of Jacobi's identity and the AGM, Trans. Amer. Math. Soc., 323 (1991), no. 2, 691-701.
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A008653, A098098, A111932, A121443, A131943, A131946, A186099.

a[ n_] := If[ n < 1, Boole[n == 0], -12 DivisorSum[ n, # Boole[ 1 == GCD[#, 6]] &]]; (* Michael Somos, Jul 07 2015 *)
a[ n_] := SeriesCoefficient[(EllipticTheta[ 4, 0, x] EllipticTheta[ 4, 0, x^3])^2 - 1/2 (EllipticTheta[ 2, 0, x^(1/2)] EllipticTheta[ 2, 0, x^(3/2)])^2, {x, 0, n}]; (* Michael Somos, Jul 07 2015 *)

{a(n) = if( n<1, n==0, -12 * sumdiv( n, d, d * (1 == gcd( d, 6) ) ) )};

{a(n) = if( n<1, n==0, -12 * direuler( p=2, n, 1 / (1 - X) / (1 - (p>3) * p * X)) [n])};

Expansion of b(x) * b(x^2) - c(x) * c(x^2) in powers of x where b(), c() are cubic AGM functions.
Expansion of (phi(-x) * phi(-x^3))^2 - 8 * x * (psi(x) * psi(x^3))^2 in powers of x where phi(), psi() are Ramanujan theta functions.
Expansion of (P(q) - 2*P(q^2) - 3*P(q^3) + 6*P(q^6)) / 2 in powers of q where P() is a Ramanujan Eisenstein series. - Michael Somos, Jul 07 2015
a(n) = -12 * A186099(n) if n>0. a(2*n) = a(n). a(2*n + 1) = - A008653(2*n + 1). a(n) = 2 * A008653(n) - A008653(2*n) = A131946(n) - 8 * A111932(n) = A131943(n) - 9 * A121443(n).
a(3*n) = a(n). a(6*n + 5) = -72 * A098098(n).- Michael Somos, Jul 07 2015

A341305 Fourier coefficients of the modular form (1/28)(E_6(t)+27E_6(3*t)).

Original entry on oeis.org

1, -18, -594, -4878, -19026, -56268, -160974, -302544, -608850, -1185858, -1856844, -2898936, -5156046, -6683292, -9983952, -15248628, -19483218, -25557444, -39133314, -44569800, -59475276, -81989424, -95664888, -115854192, -164998350, -175837518, -220548636, -288163998, -319789008, -369200700

Offset: 0

N. J. A. Sloane, Feb 13 2021

sign

Masao Koike, Modular forms on non-compact arithmetic triangle groups, Unpublished manuscript [Extensively annotated with OEIS A-numbers by N. J. A. Sloane, Feb 14 2021. I wrote 2005 on the first page but the internal evidence suggests 1997.] See page 29.

Cf. A013973.

A341305 := proc(n)
    local a;
    a := A013973(n);
    if modp(n,3) = 0 then
        a := a+27*A013973(n/3) ;
    end if;
    %/28 ;
end proc:
seq(A341305(n),n=0..10) ; # R. J. Mathar, Feb 22 2021

A013973[n_] := If[n == 0, 1, -504 DivisorSigma[5, n]];
A341305[n_] := Module[{a = A013973[n]}, If[Mod[n, 3] == 0, a = a + 27 A013973[n/3]]; a/28];
Table[A341305[n], {n, 0, 29}] (* Jean-François Alcover, Apr 30 2023, after R. J. Mathar *)

cp's OEIS Frontend

A037946 Coefficients of unique normalized cusp form Delta_22 of weight 22 for full modular group.

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A008689 Theta series of Niemeier lattice of type A_24.

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A008691 Theta series of Niemeier lattice of type A_17 E_7.

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A008695 Theta series of Niemeier lattice of type A_11 D_7 E_6.

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A008696 Theta series of Niemeier lattice of type D_6^4.

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A008700 Theta series of Niemeier lattice of type D_4^6.

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A008702 Theta series of Niemeier lattice of type A_3^8.

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A126858 Coefficients in quasimodular form F_2(q) of level 1 and weight 6.

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A341305 Fourier coefficients of the modular form (1/28)(E_6(t)+27E_6(3*t)).