A185699 - OEIS

%I #25 Mar 12 2021 22:24:46

%S 5,12,36,48,84,72,144,96,180,156,216,12,336,168,288,288,372,216,468,

%T 240,504,384,36,288,720,372,504,480,672,360,864,384,756,48,648,576,

%U 1092,456,720,672,1080,504,1152,528,84,936,864,576,1488,684,1116,864,1176,648

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

%C Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

%D B. C. Berndt, Ramanujan's Notebooks Part III, Springer-Verlag, see p. 480, Entry 8(i).

%D 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 G. C. Greubel, <a href="/A185699/b185699.txt">Table of n, a(n) for n = 0..1000</a>

%H Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>

%F 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 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 G.f. = 5 + 12*x + 36*x^2 + 48*x^3 + 84*x^4 + 72*x^5 + 144*x^6 + 96*x^7 + ...

%t terms = 54;

%t E2[x_] = 1 - 24*Sum[k*x^k/(1 - x^k), {k, 1, terms}];

%t (11*E2[x^11] - E2[x])/2 + O[x]^terms // CoefficientList[#, x]& (* _Jean-François Alcover_, Feb 27 2018 *)

%o (PARI) {a(n) = if( n<1, 5 * (n==0), 12 * (sigma( n) - if( n%11, 0, 11 * sigma( n / 11))))};

%Y Cf. A006352, A006571, A272196.

%K nonn,easy

%O 0,1

%A _Michael Somos_, Feb 10 2011