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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

G. C. Greubel, Table of n, a(n) for n = 0..10000

M. 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-Fran├žois 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

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Last modified January 19 16:32 EST 2019. Contains 319309 sequences. (Running on oeis4.)