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

0,1

Comments

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

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

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

Example

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

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 *)

Prog

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

Crossrefs

Cf. A006352, A006571, A272196.

Sequence in context: A032281 A294654 A229043 * A301873 A368094 A077918

Adjacent sequences: A185696 A185697 A185698 * A185700 A185701 A185702

Keyword

nonn,easy

Author

Michael Somos, Feb 10 2011

Status

approved