A282012
Coefficients in q-expansion of E_4^4, where E_4 is the Eisenstein series shown in A004009.
Original entry on oeis.org
1, 960, 354240, 61543680, 4858169280, 137745912960, 2120861041920, 21423820362240, 158753769048000, 928983317334720, 4512174992346240, 18847874280625920, 69518972236842240, 230951926208599680, 701949379778818560, 1975788826748167680
Offset: 0
- G. E. Andrews and B. C. Berndt, Ramanujan's lost notebook, Part III, Springer, New York, 2012, See p. 207.
-
terms = 16;
E4[x_] = 1 + 240*Sum[k^3*x^k/(1 - x^k), {k, 1, terms}];
E4[x]^4 + 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, 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, 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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