A281979 -id:A281979 - cp's OEIS frontend

A027860 a(n) = (-tau(n) + sigma_11(n)) / 691, where tau is Ramanujan's tau (A000594), sigma_11(n) = Sum_{ d divides n } d^11 (A013959).

0, 3, 256, 6075, 70656, 525300, 2861568, 12437115, 45414400, 144788634, 412896000, 1075797268, 2593575936, 5863302600, 12517805568, 25471460475, 49597544448, 93053764671, 168582124800, 296526859818, 506916761600, 846025507836, 1378885295616, 2203231674900

Offset: 1

Bill Gosper

nonn

It appears that this sequence is strictly increasing. - Jianing Song, Aug 05 2018

"Number Theory I", vol. 49 of the Encyc. of Math. Sci.

Robert Israel, Table of n, a(n) for n = 1..2000

Cf. A000594, A013959.

Similar sequences: A281788, A281876, A281928, A281956, A281979.

(sum(n^11*q^n/(1-q^n), n,1,inf)-q*prod(1-q^n,n,1,inf)^24)/691; taylor(%,q,0,24);

N:= 100: # to get a(1) to a(N)
S:= series(q*mul((1-q^k)^24,k=1..N),q,N+1):
seq((-coeff(S,q,n) + add(d^11, d = numtheory:-divisors(n)))/691, n=1..N); # Robert Israel, Nov 12 2014

{0}~Join~Array[(-RamanujanTau@ # + DivisorSigma[11, #])/691 &, 24] (* Michael De Vlieger, Aug 05 2018 *)

a(n) = (sigma(n, 11) - polcoeff( x * eta(x + x * O(x^n))^24, n))/691; \\ for n>0; Michel Marcus, Nov 12 2014

def A027860List(len):
    r = list(delta_qexp(len+1))
    return [(sigma(n, 11) - r[n])/691 for n in (1..len)]
A027860List(24) # Peter Luschny, Aug 20 2018

a(n) = (A013959(n) - A000594(n))/691. - Michel Marcus, Nov 12 2014

More terms from Michel Marcus, Nov 12 2014

A282048 Coefficients in q-expansion of E_4^5*E_6, where E_4 and E_6 are respectively the Eisenstein series A004009 and A013973.

Original entry on oeis.org

1, 696, -34632, -167186976, -64422848328, -11387712944304, -1037073232984608, -48892286706157632, -1378097272692189000, -26188038166214133672, -364779879415169299632, -3952277018332870144608, -34798618196377082329632, -257403706082325167732976

Offset: 0

Seiichi Manyama, Feb 05 2017

sign

G. E. Andrews and B. C. Berndt, Ramanujan's lost notebook, Part III, Springer, New York, 2012, See p. 208.

Seiichi Manyama, Table of n, a(n) for n = 0..1000

Cf. A004009 (E_4), A013973 (E_6), A013974 (E_4*E_6 = E_10), A058550 (E_4^2*E_6 = E_14), A282000 (E_4^3*E_6), A282047 (E_4^4*E_6), this sequence (E_4^5*E_6).

Cf. A037947, A281959, A281979.

terms = 14;
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]^5*E6[x] + O[x]^terms // CoefficientList[#, x]& (* Jean-François Alcover, Feb 27 2018 *)

-24 * A281959(n) = 657931 * a(n) - 457920000 * A037947(n) for n > 0.

cp's OEIS Frontend

A027860 a(n) = (-tau(n) + sigma_11(n)) / 691, where tau is Ramanujan's tau (A000594), sigma_11(n) = Sum_{ d divides n } d^11 (A013959).

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