A282596
Coefficients in q-expansion of E_2*E_4^2*E_6, where E_2, E_4, E_6 are the Eisenstein series shown in A006352, A004009, A013973, respectively.
Original entry on oeis.org
1, -48, -196128, -33542976, -678319104, 12136422240, 509314518144, 7469015889792, 68272650653760, 458377820557584, 2454769903187520, 11035857376010304, 43103740076823552, 149954656815201504, 473331019057949952, 1375248429330791040, 3719662610125117632
Offset: 0
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terms = 17;
E2[x_] = 1 - 24*Sum[k*x^k/(1 - x^k), {k, 1, terms}];
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}];
E2[x]* E4[x]^2 *E6[x] + O[x]^terms // CoefficientList[#, x]& (* Jean-François Alcover, Feb 27 2018 *)
A282792
Coefficients in q-expansion of E_2^2*E_4*E_6, where E_2, E_4, E_6 are the Eisenstein series shown in A006352, A004009, A013973, respectively.
Original entry on oeis.org
1, -312, -122328, 1193376, 120735336, 123318576, -26119268064, -383848045248, -3132125965080, -18290795499096, -84925855577232, -331983655889184, -1133781877844448, -3470165144567184, -9697162366507968, -25093220330304576, -60786860467926552
Offset: 0
Cf.
A282102 (E_2*E_4*E_6), this sequence (E_2^2*E_4*E_6),
A282596 (E_2*E_4^2*E_6),
A282547 (E_2*E_4*E_6^2).
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terms = 17;
E2[x_] = 1 - 24*Sum[k*x^k/(1 - x^k), {k, 1, terms}];
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]^2*E6[x]*E6[x] + O[x]^terms // CoefficientList[#, x]& (* Jean-François Alcover, Feb 27 2018 *)
A282777
Expansion of phi_{16, 1}(x) where phi_{r, s}(x) = Sum_{n, m>0} m^r * n^s * x^{m*n}.
Original entry on oeis.org
0, 1, 65538, 43046724, 4295098372, 152587890630, 2821196197512, 33232930569608, 281483566907400, 1853020317992013, 10000305176108940, 45949729863572172, 184889914172333328, 665416609183179854, 2178019803670969104, 6568408813691796120
Offset: 0
- George E. Andrews and Bruce C. Berndt, Ramanujan's lost notebook, Part III, Springer, New York, 2012. See p. 212.
Cf.
A064987 (phi_{2, 1}),
A281372 (phi_{4, 1}),
A282050 (phi_{6, 1}),
A282060 (phi_{8, 1}),
A282254 (phi_{10, 1}),
A282548 (phi_{12, 1}),
A282597 (phi_{14, 1}), this sequence (phi_{16, 1}).
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Table[If[n==0, 0, n * DivisorSigma[15, n]], {n, 0, 15}] (* Indranil Ghosh, Mar 11 2017 *)
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for(n=0, 15, print1(if(n==0, 0, n * sigma(n, 15)), ", ")) \\ Indranil Ghosh, Mar 11 2017
Showing 1-3 of 3 results.
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