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

5, 12, 36, 48, 84, 72, 144, 96, 180, 156, 216, 12, 336, 168, 288, 288, 372, 216, 468, 240, 504, 384, 36, 288, 720, 372, 504, 480, 672, 360, 864, 384, 756, 48, 648, 576, 1092, 456, 720, 672, 1080, 504, 1152, 528, 84, 936, 864, 576, 1488, 684, 1116, 864, 1176, 648

Offset: 0

Author

Michael Somos, Feb 10 2011

Keywords

• nonn
• easy

Comments

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

Examples

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

References

• B. C. Berndt, Ramanujan's Notebooks Part III, Springer-Verlag, see p. 480, Entry 8(i).
• Carlos J. Moreno and Samuel S. Wagstaff, Jr., Sums of Squares of Integers, Chapman & Hall/CRC, Boca Raton, London, New York, p. 246 (corrected).

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Michael Somos, Introduction to Ramanujan theta functions
• Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Crossrefs

Cf. A006352, A006571, A272196.

Programs

Mathematica

terms = 54;
E2[x_] = 1 - 24*Sum[k*x^k/(1 - x^k), {k, 1, terms}];
(11*E2[x^11] - E2[x])/2 + O[x]^terms // CoefficientList[#, x]& (* Jean-François Alcover, Feb 27 2018 *)

PARI

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

Formula

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.

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