A005931 Theta series of the coset of the E_7 lattice in its dual.

56, 576, 1512, 4032, 5544, 12096, 13664, 24192, 27216, 44352, 41832, 72576, 67536, 100800, 101304, 145728, 126504, 205632, 176456, 249984, 234360, 326592, 277200, 423360, 355320, 479808, 439992, 608832, 494928, 749952, 599760, 806400, 745416

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn

Comments

Ramanujan theta functions: f(q) := Prod_{k>=1} (1-(-q)^k) (see A121373), phi(q) := theta_3(q) := Sum_{k=-oo..oo} q^(k^2) (A000122), psi(q) := Sum_{k=0..oo} q^(k*(k+1)/2) (A010054), chi(q) := Prod_{k>=0} (1+q^(2k+1)) (A000700).

Examples

			56*q^(3/2) + 576*q^(7/2) + 1512*q^(11/2) + 4032*q^(15/2) + 5544*q^(19/2) + ...
		

References

• J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 125. Equation (113)
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• G. C. Greubel, Table of n, a(n) for n = 0..10000
• Michael Somos, Introduction to Ramanujan theta functions
• Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Programs

Mathematica

terms = 33; phi[q_] := EllipticTheta[3, 0, q]; chi[q_] := ((1 - InverseEllipticNomeQ[q])*InverseEllipticNomeQ[q]/(16*q))^(-1/24); psi[q_] := (1/2)*q^(-1/8)*EllipticTheta[2, 0, q^(1/2)]; s = 56*psi[q^2]^3 * phi[q]^4 + 128*q*psi[q^2]^7 + O[q]^terms; CoefficientList[s, q] (* Jean-François Alcover, Jul 04 2017, after Michael Somos *)

PARI

{a(n)= local(A, B); if(n<0, 0, n++; A= sum(k=1, sqrtint(n), 2*x^k^2, 1+x*O(x^n)); B= subst(A,x,-x); polcoeff( (A^4 -B^4)* (8*A^4 -B^4)/ 2/ sum(k=0, sqrtint( 4*n+1)\2, x^(k^2+k), x*O(x^n)), n))} /* Michael Somos, Jun 11 2007*/

Formula

Expansion of 56* psi(q^2)^3* phi(q)^4 +128* q* psi(q^2)^7 in powers of q where phi(),psi() are Ramanujan theta functions. - Michael Somos, Jun 11 2007