A005931 Theta series of the coset of the E_7 lattice in its dual. (Formerly M5313)

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

0,1

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).

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

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

Example

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

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 *)

Prog

(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*/

Crossrefs

Sequence in context: A189497 A219712 A244429 * A027793 A244945 A055747

Adjacent sequences: A005928 A005929 A005930 * A005932 A005933 A005934

Keyword

nonn

Author

N. J. A. Sloane

Status

approved