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%I #55 Jul 14 2023 17:09:35
%S 1,216,-3348,13888,52110,-723168,2822456,-4078080,-3139803,11255760,
%T 20586852,-46497024,-190073338,609650496,-174464280,-1335947264,
%U 1646527986,-678197448,1563257180,723703680,-9449582688,4446760032,9451116072,13653411840,-27802126025,-41055841008
%N Coefficients of unique normalized cusp form Delta_16 of weight 16 for full modular group.
%H Seiichi Manyama, <a href="/A027364/b027364.txt">Table of n, a(n) for n = 1..1000</a>
%H Steven R. Finch, <a href="/A000521/a000521_1.pdf">Modular forms on SL_2(Z)</a>, December 28, 2005. [Cached copy, with permission of the author]
%H Author?, <a href="http://www.math.okstate.edu/~loriw/degree2/degree2hm/level1/weight16/c16.html">Table of coefficients c16(n) of the weight 16 cusp form on Gamma_0(1) for n up to 1000</a> [Broken link]
%H F. Q. Gouvea, <a href="http://projecteuclid.org/euclid.em/1047920420">Non-ordinary primes</a>, Experimental Mathematics 6 195, 1997.
%H S. C. Milne, <a href="https://arxiv.org/abs/math/0009130">Hankel determinants of Eisenstein series</a>, preprint, arXiv:0009130 [math.NT], 2000.
%H H. P. F. Swinnerton-Dyer, <a href="https://doi.org/10.1007/978-3-540-37802-0_1">On l-adic representations and congruences for coefficients of modular forms</a>, pp. 1-55 of Modular Functions of One Variable III (Antwerp 1972), Lect. Notes Math., 350, 1973.
%H <a href="/index/Gre#groups_modular">Index entries for sequences related to modular groups</a>
%F 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).
%F (E_4(q)^4 - E_6(q)^2*E_4(q))/1728.
%F a(n) == A013963(n) mod 3617. - _Seiichi Manyama_, Feb 01 2017
%F G.f.: -691/(1728*250) * (E_4(q)*E_12(q) - E_8(q)^2). - _Seiichi Manyama_, Jul 25 2017
%e G.f. = q + 216*q^2 - 3348*q^3 + 13888*q^4 + 52110*q^5 - 723168*q^6 + ...
%p 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
%t terms = 26;
%t E4[x_] = 1 + 240*Sum[k^3*x^k/(1 - x^k), {k, 1, terms+1}];
%t E6[x_] = 1 - 504*Sum[k^5*x^k/(1 - x^k), {k, 1, terms+1}];
%t (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_ *)
%o (PARI) 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
%Y Cf. A013963, A126763, A290152, A290178.
%Y The unique normalized cusp form: A000594 (k=12), this sequence (k=16), A037944 (k=18), A037945 (k=20), A037946 (k=22), A037947 (k=26).
%K sign,easy
%O 1,2
%A Paolo Dominici (pl.dm(AT)libero.it), _N. J. A. Sloane_
%E More terms from C. Ronaldo (aga_new_ac(AT)hotmail.com), Jan 17 2005
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