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A005931 Theta series of the coset of the E_7 lattice in its dual.
(Formerly M5313)
3
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 (list; graph; refs; listen; history; text; internal format)
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
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
KEYWORD
nonn
AUTHOR
STATUS
approved

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Last modified March 1 20:05 EST 2024. Contains 370443 sequences. (Running on oeis4.)