Original entry on oeis.org
0, 3, 2944, 391635, 17392128, 385866060, 5303086848, 51332824275, 380176030720, 2279635315794, 11522261136000, 50576242992268, 197196432781824, 695091512105880, 2246019242126592, 6728295917456595, 18857917384178688, 49830812542200039
Offset: 1
a(1) = (1 - 1)/43867 = 0.
a(2) = (131073 - (-528))/43867 = 3.
a(3) = (129140164 - (-4284))/43867 = 2944.
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
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
-
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
More terms from C. Ronaldo (aga_new_ac(AT)hotmail.com), Jan 17 2005
A282000
Coefficients in q-expansion of E_4^3*E_6, where E_4 and E_6 are respectively the Eisenstein series A004009 and A013973.
Original entry on oeis.org
1, 216, -200232, -85500576, -11218984488, -499862636784, -11084671590048, -152346382155072, -1474691273530920, -10921720940625672, -65489246355989232, -331011680696545248, -1452954445366288032, -5665058572086302256, -19968589327695656256
Offset: 0
- G. E. Andrews and B. C. Berndt, Ramanujan's lost notebook, Part III, Springer, New York, 2012, See p. 208.
-
terms = 15;
E4[x_] = 1 + 240*Sum[k^3*x^k/(1 - x^k), {k, 1, terms}];
E6[x_] = 1 - 504*Sum[k^5*x^k/(1 - x^k), {k, 1, terms}];
E4[x]^3*E6[x] + O[x]^terms // CoefficientList[#, x]& (* Jean-François Alcover, Feb 26 2018 *)
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