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A185699 Expansion of (11 * E_2(x^11) - E_2(x)) / 2 in powers of x where E_2() is an Eisenstein series.

%I A185699 #26 Feb 16 2025 08:33:13
%S A185699 5,12,36,48,84,72,144,96,180,156,216,12,336,168,288,288,372,216,468,
%T A185699 240,504,384,36,288,720,372,504,480,672,360,864,384,756,48,648,576,
%U A185699 1092,456,720,672,1080,504,1152,528,84,936,864,576,1488,684,1116,864,1176,648
%N A185699 Expansion of (11 * E_2(x^11) - E_2(x)) / 2 in powers of x where E_2() is an Eisenstein series.
%C A185699 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%D A185699 B. C. Berndt, Ramanujan's Notebooks Part III, Springer-Verlag, see p. 480, Entry 8(i).
%D A185699 Carlos J. Moreno and Samuel S. Wagstaff, Jr., Sums of Squares of Integers, Chapman & Hall/CRC, Boca Raton, London, New York,  p. 246 (corrected).
%H A185699 G. C. Greubel, <a href="/A185699/b185699.txt">Table of n, a(n) for n = 0..1000</a>
%H A185699 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A185699 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A185699 Expansion of 5 * (phi(x) * phi(x^11))^2 - 20 * x * (f(x) * f(x^11))^2 + 32 * x^2 * (f(-x^2) * f(-x^22))^2 - 20 * x^3 * (psi(-x) * psi(-x^11))^2 in powers of x where f(), phi(), psi() are Ramanujan theta functions.
%F A185699 Expansion of 5 + 12*Sum_{n>=1} Chi0(n)*n*q^n / (1 - q^n), where Chi0(n) = 1 if gcd(n,11) = 1 and 0 otherwise. See the Moreno-Wagstaff reference p. 246, second equation multiplied by 12 (a misprint has been corrected, after mail exchange with C. J. Moreno). - _Wolfdieter Lang_, Jan 02 2017
%e A185699 G.f. = 5 + 12*x + 36*x^2 + 48*x^3 + 84*x^4 + 72*x^5 + 144*x^6 + 96*x^7 + ...
%t A185699 terms = 54;
%t A185699 E2[x_] = 1 - 24*Sum[k*x^k/(1 - x^k), {k, 1, terms}];
%t A185699 (11*E2[x^11] - E2[x])/2 + O[x]^terms // CoefficientList[#, x]& (* _Jean-François Alcover_, Feb 27 2018 *)
%o A185699 (PARI) {a(n) = if( n<1, 5 * (n==0), 12 * (sigma( n) - if( n%11, 0, 11 * sigma( n / 11))))};
%Y A185699 Cf. A006352, A006571, A272196.
%K A185699 nonn,easy
%O A185699 0,1
%A A185699 _Michael Somos_, Feb 10 2011