A282015
Coefficients in q-expansion of E_4^5, where E_4 is the Eisenstein series shown in A004009.
Original entry on oeis.org
1, 1200, 586800, 148641600, 20400279600, 1439038231200, 46093334702400, 861697555612800, 10894180752126000, 102121497049868400, 755966260027216800, 4623420005167550400, 24151632380348692800, 110516281318431693600, 451789183426135939200
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}];
E4[x]^5 + O[x]^terms // CoefficientList[#, x]& (* Jean-François Alcover, Feb 26 2018 *)
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.
-
(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
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{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
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, 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.
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.
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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