A290048 -id:A290048 - cp's OEIS frontend

A290178 Coefficients in expansion of E_4*Delta^2 where Delta is the generating function of Ramanujan's tau function (A000594).

1, 192, -8280, 147200, -1438020, 7491456, -4626880, -246965760, 2112385950, -9443825600, 23625035616, -14413771008, -118710609640, 427914230400, -467038103040, -645319017984, 1640006523477, 2800373100480, -8506579320400, -21655683517440, 108181106829972

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: this sequence (k=4), A290048 (k=6), A290180 (k=8), A290181 (k=10), A290182 (k=14).

Cf. A000594, A004009 (E_4), A290152.

terms = 21;
E4[x_] = 1 + 240*Sum[k^3*x^k/(1 - x^k), {k, 1, terms}];
E4[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_4, E_8, E_10 ; E_6, E_10, E_12 ; E_8, E_12, E_14]. G.f. is 691^2*b(q)/(1728^2*21^2*250).

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

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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), 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).

A037944 Coefficients of unique normalized cusp form Delta_18 of weight 18 for full modular group.

Original entry on oeis.org

1, -528, -4284, 147712, -1025850, 2261952, 3225992, -8785920, -110787507, 541648800, -753618228, -632798208, 2541064526, -1703323776, 4394741400, -14721941504, -5429742318, 58495803696, 1487499860, -151530355200

Offset: 1

N. J. A. Sloane

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			G.f. = q - 528*q^2 - 4284*q^3 + 147712*q^4 - 1025850*q^5 + 2261952*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]
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.
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. A000594, A013965, A013973, A027364, A037945, A037946, A037947, A290048.

terms = 20;
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}];
E12[x_] = 1 + (65520/691)*Sum[k^11*x^k/(1 - x^k), {k, 1, terms}];
E14[x_] = 1 - 24*Sum[k^13*x^k/(1 - x^k), {k, 1, terms}];
(691/(1728*250))*(E4[x]*E14[x] - E6[x]*E12[x]) + O[x]^(terms+1) // CoefficientList[#, x]& // Rest (* Jean-François Alcover, Feb 27 2018, after Seiichi Manyama *)

{a(n) = if( n<0, 0, polcoeff( x * eta(x + x * O(x^n))^24 * (1 - 504 * sum( k=1, n, sigma( k, 5) * x^k)), n))}; /* Michael Somos, Mar 18 2012 */

Convolution product of A000594 and A013973. - Michael Somos, Mar 18 2012
a(n) == A013965(n) mod 43867. - Seiichi Manyama, Feb 02 2017
G.f.: 691/(1728*250) * (E_4(q)*E_14(q) - E_6(q)*E_12(q)). - Seiichi Manyama, Jul 25 2017

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).

cp's OEIS Frontend

A290178 Coefficients in expansion of E_4*Delta^2 where Delta is the generating function of Ramanujan's tau function (A000594).

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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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A037944 Coefficients of unique normalized cusp form Delta_18 of weight 18 for full modular group.

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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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