A290178 -id:A290178 - cp's OEIS frontend

A290048 Coefficients in expansion of E_6*Delta^2 where Delta is the generating function of Ramanujan's tau function (A000594).

1, -552, 8640, 116000, -4868460, 67855536, -544522240, 2742137280, -8237774250, 10592091400, 3366617856, 113971542048, -1217020425880, 4535746506000, -5415752171520, -19090509870144, 93580817811453, -142801363479240, -80721277168000, 665065363025280

Offset: 2

Seiichi Manyama, Jul 19 2017

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Seiichi Manyama, Table of n, a(n) for n = 2..1000
S. C. Milne, Hankel determinants of Eisenstein series, preprint, arXiv:0009130 [math.NT], 2000.

Cf. A000594, A010839, A013973 (E_6).
Cf. A282382, A282461 (E_6*E_10*E_14 = E_10^3), A290049, A290050.
E_k*Delta^2: A290178 (k=4), this sequence (k=6), A290180 (k=8), A290181 (k=10), A290182 (k=14).

terms = 20;
E6[x_] = 1 - 504*Sum[k^5*x^k/(1 - x^k), {k, 1, terms}];
E6[x]*QPochhammer[x]^48 + O[x]^terms // CoefficientList[#, x]& (* Jean-François Alcover, Feb 26 2018 *)

Let b(q) be the determinant of the 3 X 3 Hankel matrix [E_6, E_8, E_10 ; E_8, E_10, E_12 ; E_10, E_12, E_14]. G.f. is -691^2*b(q)/(1728^2*250^2).

a(n) = (A290050(n) - 2*691*A290049(n) + 691^2*A282382(n))/(1728^2*250^2).

A027364 Coefficients of unique normalized cusp form Delta_16 of weight 16 for full modular group.

Original entry on oeis.org

1, 216, -3348, 13888, 52110, -723168, 2822456, -4078080, -3139803, 11255760, 20586852, -46497024, -190073338, 609650496, -174464280, -1335947264, 1646527986, -678197448, 1563257180, 723703680, -9449582688, 4446760032, 9451116072, 13653411840, -27802126025, -41055841008

Offset: 1

Paolo Dominici (pl.dm(AT)libero.it), N. J. A. Sloane

			G.f. = q + 216*q^2 - 3348*q^3 + 13888*q^4 + 52110*q^5 - 723168*q^6 + ...

Seiichi Manyama, Table of n, a(n) for n = 1..1000
Steven R. Finch, Modular forms on SL_2(Z), December 28, 2005. [Cached copy, with permission of the author]
Author?, Table of coefficients c16(n) of the weight 16 cusp form on Gamma_0(1) for n up to 1000
F. Q. Gouvea, Non-ordinary primes, Experimental Mathematics 6 195, 1997.
LMFDB, Newform orbit 1.16.a.a.
S. C. Milne, Hankel determinants of Eisenstein series, preprint, arXiv:0009130 [math.NT], 2000.
H. P. F. Swinnerton-Dyer, On l-adic representations and congruences for coefficients of modular forms, pp. 1-55 of Modular Functions of One Variable III (Antwerp 1972), Lect. Notes Math., 350, 1973.
Index entries for sequences related to modular groups

Cf. A013963, A126763, A290152, A290178.

The unique normalized cusp form: A000594 (k=12), this sequence (k=16), A037944 (k=18), A037945 (k=20), A037946 (k=22), A037947 (k=26).

with(numtheory): DO := qs -> q*diff(qs,q)/2: E2:=1-24*add(sigma(n)*q^(2*n),n=1..100): delta16:=(-1/24)*(DO@@6)(E2)*E2+(9/8)*(DO@@5)(E2)*(DO@@1)(E2)-(45/8)*(DO@@4)(E2)*(DO@@2)(E2)+(55/12)*(DO@@3)(E2)*(DO@@3)(E2):seq(coeff(delta16,q,2*i),i=1..40); with(numtheory): E2n:=n->1-(4*n/bernoulli(2*n))*add(sigma[2*n-1](k)*q^(2*k),k=1..100): qs:=(E2n(2)^4-E2n(3)^2*E2n(2))/1728: seq(coeff(qs,q,2*i),i=1..40); # C. Ronaldo

terms = 26;
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]^4 - E6[x]^2*E4[x])/1728 + O[x]^(terms+1) // CoefficientList[#, x]& // Rest (* Jean-François Alcover, Feb 27 2018, after Seiichi Manyama *)

N=66; q='q+O('q^N); Vec(q*(1+240*sum(n=1,N,sigma(n,3)*q^n))*eta(q)^24) \\ Joerg Arndt, Nov 23 2015

G.f.: q*(1 + 240*Sum_{n>=1} sigma_3(n)q^n) Product_{k>=1} (1-q^k)^24, where sigma_3(n) is the sum of the cubes of the divisors of n (A001158).
(E_4(q)^4 - E_6(q)^2*E_4(q))/1728.
a(n) == A013963(n) mod 3617. - Seiichi Manyama, Feb 01 2017
G.f.: -691/(1728*250) * (E_4(q)*E_12(q) - E_8(q)^2). - Seiichi Manyama, Jul 25 2017

More terms from C. Ronaldo (aga_new_ac(AT)hotmail.com), Jan 17 2005

A290180 Coefficients in expansion of E_8*Delta^2 where Delta is the generating function of Ramanujan's tau function (A000594).

Original entry on oeis.org

1, 432, 39960, -1418560, 17312940, -71928864, -462815680, 7500885120, -38038437810, 29000909200, 729783353376, -4661016429888, 13691625085880, -16503845217120, -14982974507520, 45085348093056, 99234456545637, -157805792764560, -1644659689877680

Offset: 2

Seiichi Manyama, Jul 23 2017

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Seiichi Manyama, Table of n, a(n) for n = 2..1000
S. C. Milne, Hankel determinants of Eisenstein series, preprint, arXiv:0009130 [math.NT], 2000.

E_k*Delta^2: A290178 (k=4), A290048 (k=6), this sequence (k=8), A290181 (k=10), A290182 (k=14).

Cf. A000594, A008410 (E_8).

terms = 19;
E8[x_] = 1 + 480*Sum[k^7*x^k/(1 - x^k), {k, 1, terms}];
E8[x]*QPochhammer[x]^48 + O[x]^terms // CoefficientList[#, x]& (* Jean-François Alcover, Feb 26 2018 *)

Let b(q) be the determinant of the 3 X 3 matrix [E_6, E_8, E_12 ; E_8, E_10, E_14 ; E_10, E_12, E_16]. G.f. is -691^2*3617*b(q)/(1728^2*2^3*3*5^3*7^2*467).

A290181 Coefficients in expansion of E_10*Delta^2 where Delta is the generating function of Ramanujan's tau function (A000594).

Original entry on oeis.org

1, -312, -121680, 1004000, 37942020, -801594864, 6139193600, -11831002560, -151614128250, 1346611783000, -4592794000704, 3738595861728, 15192491492360, 47281379454000, -737660590018560, 2662090686805056, -3290770281735027, -4884703150768920

Offset: 2

Seiichi Manyama, Jul 23 2017

sign

Seiichi Manyama, Table of n, a(n) for n = 2..1000
S. C. Milne, Hankel determinants of Eisenstein series, preprint, arXiv:0009130 [math.NT], 2000.

E_k*Delta^2: A290178 (k=4), A290048 (k=6), A290180 (k=8), this sequence (k=10), A290182 (k=14).

Cf. A000594, A013974 (E_10).

terms = 18;
E10[x_] = 1 - 264*Sum[k^9*x^k/(1 - x^k), {k, 1, terms}];
E10[x]*QPochhammer[x]^48 + O[x]^terms // CoefficientList[#, x]& (* Jean-François Alcover, Feb 26 2018 *)

Let b(q) be the determinant of the 3 X 3 matrix [E_6, E_10, E_12 ; E_8, E_12, E_14 ; E_10, E_14, E_16]. G.f. is 691^2*3617*b(q)/(1728^2*2^2*3*5^6*7^2*13).

A290182 Coefficients in expansion of E_14*Delta^2 where Delta is the generating function of Ramanujan's tau function (A000594).

Original entry on oeis.org

1, -72, -194400, -28866400, 13994100, 9650004336, -99683138560, -1007380800, 5570606272950, -32186306471000, -2717893793664, 724443400725408, -2662202398202200, -401005712372400, 19385312101171200, 24633489938571456, -449375771787124707

Offset: 2

Seiichi Manyama, Jul 23 2017

sign

Seiichi Manyama, Table of n, a(n) for n = 2..1000
S. C. Milne, Hankel determinants of Eisenstein series, preprint, arXiv:0009130 [math.NT], 2000.

E_k*Delta^2: A290178 (k=4), A290048 (k=6), A290180 (k=8), A290181 (k=10), this sequence (k=14).

Cf. A000594, A058550 (E_14).

terms = 17;
E14[x_] = 1 - 24*Sum[k^13*x^k/(1 - x^k), {k, 1, terms}];
E14[x]*QPochhammer[x]^48 + O[x]^terms // CoefficientList[#, x]& (* Jean-François Alcover, Feb 26 2018 *)

Let b(q) be the determinant of the 3 X 3 matrix [E_8, E_10, E_14 ; E_10, E_12, E_16 ; E_12, E_14, E_18]. G.f. is -691^2*3617*43867*b(q)/(1728^2*2^6*3*5^3*7^2*97*7213).

A290152 Coefficients in expansion of E_4*Delta^3 where Delta is the generating function of Ramanujan's tau function (A000594).

Original entry on oeis.org

1, 168, -12636, 392832, -7335174, 92207808, -804651624, 4626614784, -11834988165, -73870961696, 1115908456740, -7498139072256, 32630722986078, -90379426346496, 94395618447768, 450271639673856, -2625847472007243, 6203580643521072, -3151366507609936

Offset: 3

Seiichi Manyama, Jul 21 2017

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Seiichi Manyama, Table of n, a(n) for n = 3..1000
S. C. Milne, Hankel determinants of Eisenstein series, preprint, arXiv:0009130 [math.NT], 2000.

Cf. A000594, A004009 (E_4).

Cf. A290048, A290178.

terms = 19;
E4[x_] = 1 + 240*Sum[k^3*x^k/(1 - x^k), {k, 1, terms}];
E4[x]*QPochhammer[x]^(3*24) + O[x]^terms // CoefficientList[#, x]& (* Jean-François Alcover, Feb 26 2018 *)

Let b(q) be the determinant of the 4 X 4 Hankel matrix [E_4, E_6, E_8, E_10 ; E_6, E_8, E_10, E_12 ; E_8, E_10, E_12, E_14 ; E_10, E_12, E_14, E_16]. G.f. is -691^3*3617*b(q)/(1728^3*2^4*3*5^6*7^2*467).

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A290048 Coefficients in expansion of E_6*Delta^2 where Delta is the generating function of Ramanujan's tau function (A000594).

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A027364 Coefficients of unique normalized cusp form Delta_16 of weight 16 for full modular group.

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A290180 Coefficients in expansion of E_8*Delta^2 where Delta is the generating function of Ramanujan's tau function (A000594).

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A290181 Coefficients in expansion of E_10*Delta^2 where Delta is the generating function of Ramanujan's tau function (A000594).

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A290182 Coefficients in expansion of E_14*Delta^2 where Delta is the generating function of Ramanujan's tau function (A000594).

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A290152 Coefficients in expansion of E_4*Delta^3 where Delta is the generating function of Ramanujan's tau function (A000594).

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