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).
Original entry on oeis.org
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
- "Number Theory I", vol. 49 of the Encyc. of Math. Sci.
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(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);
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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
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{0}~Join~Array[(-RamanujanTau@ # + DivisorSigma[11, #])/691 &, 24] (* Michael De Vlieger, Aug 05 2018 *)
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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
Original entry on oeis.org
0, 3, 6656, 1574235, 109234176, 3489819540, 65281655808, 825351571995, 7736349470720, 57270269768634, 350259092774400, 1829670576438068, 8372440970643456, 34226453991167880, 126958657929489408, 432721923827171355, 1369171676955783168, 4056082931864408991, 11330441127202890240, 30026115193307387658, 75874353000273633280, 183636989491548765276
Offset: 1
a(1) = (1 - 1)/174611 = 0.
a(2) = (524289 - 456)/174611 = 3.
a(3) = (1162261468 - 50652)/174611 = 6656.
Original entry on oeis.org
0, 9, 3968, 296865, 8437248, 129997260, 1312568064, 9727799265, 56923182080, 276480648702, 1154893046400, 4259743681004, 14151477247488, 43011568291320, 121065502097664, 318760489739745, 791380439553024, 1865315725321293, 4197159808767360, 9059718006875214
Offset: 1
a(1) = (1 - 1)/3617 = 0.
a(2) = (32769 - 216)/3617 = 9.
a(3) = (14348908 - (-3348))/3617 = 3968.
Original entry on oeis.org
0, 27, 134656, 56615355, 6138243072, 282390755580, 7190065585152, 118730950577595, 1408531971420160, 12872835457479666, 95262154452748800, 592216338844654972, 3180419513581234176, 15078667591360144440, 64208193499209765888, 248996850497620053435
Offset: 1
a(1) = (1 - 1)/77683 = 0.
a(2) = (2097153 - (-288))/77683 = 27.
a(3) = (10460353204 - (-128844))/77683 = 134656.
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.
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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 *)
Original entry on oeis.org
0, 51, 1287808, 1711273635, 452970333696, 43211657266860, 2038311950075136, 57420813107839395, 1091144797392901120, 15199162675148592018, 164678453263146595200, 1449942615368630353516, 10725152052216567264768, 68394401763888606334680
Offset: 1
a(1) = (1 - 1)/657931 = 0.
a(2) = (33554433 - (-48))/657931 = 51.
a(3) = (847288609444 - (-195804))/657931 = 1287808.
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