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A027364
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Coefficients of unique normalized cusp form Delta_16 of weight 16 for full modular group.
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6
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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
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OFFSET
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1,2
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LINKS
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FORMULA
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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.
G.f.: -691/(1728*250) * (E_4(q)*E_12(q) - E_8(q)^2). - Seiichi Manyama, Jul 25 2017
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EXAMPLE
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G.f. = q + 216*q^2 - 3348*q^3 + 13888*q^4 + 52110*q^5 - 723168*q^6 + ...
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MAPLE
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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
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MATHEMATICA
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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}];
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PROG
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(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
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CROSSREFS
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KEYWORD
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sign,easy
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AUTHOR
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EXTENSIONS
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More terms from C. Ronaldo (aga_new_ac(AT)hotmail.com), Jan 17 2005
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STATUS
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approved
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