OFFSET
0,2
COMMENTS
LINKS
Seiichi Manyama, 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
K. S. Williams, The parents of Jacobi's four squares theorem are unique, Amer. Math. Monthly, 120 (2013), 329-345.
FORMULA
a(n) = -8*sigma(n) + 48*sigma(n/2) - 64*sigma(n/4) for n>0, where sigma(n) = A000203(n) if n is an integer, otherwise 0.
Euler transform of period 2 sequence [ -8, -4, ...].
G.f.: Prod_{k>0} (1 - x^k)^8 / (1 - x^(2k))^4 = 1 + Sum_{k>0} k * (-8 * x^k / (1 - x^k) + 48 * x^(2*k) /(1 - x^(2*k)) - 64 * x^(4*k)/(1 - x^(4*k))).
G.f. theta_4(q)^4 = (Sum_{k} (-q)^(k^2))^4.
Expansion of phi(-q)^4 in powers of q where phi() is a Ramanujan theta function. - Michael Somos, Nov 01 2006
G.f. A(x) satisfies 0 = f(A(x), A(x^3), A(x^9)) where f(u, v, w) = v^4 - 30*u*v^2*w + 12*u*v*w * (u + 9*w) - u*w * (u^2 + 9*w*u + 81*w^2).
a(n) = (-1)^n * A000118(n). a(n) = 8 * A109506(n) unless n=0. a(2*n) = A004011(n). a(2*n + 1) = -A005879(n).
a(0) = 1, a(n) = -(8/n)*Sum_{k=1..n} A002131(k)*a(n-k) for n > 0. - Seiichi Manyama, May 02 2017
EXAMPLE
G.f. = 1 - 8*q + 24*q^2 - 32*q^3 + 24*q^4 - 48*q^5 + 96*q^6 - 64*q^7 + 24*q^8 - ...
MATHEMATICA
CoefficientList[ Series[1 + Sum[k(-8x^k/(1 - x^k) + 48x^(2k)/(1 - x^(2k)) - 64x^(4k)/(1 - x^(4k))), {k, 1, 60}], {x, 0, 60}], x] (* Robert G. Wilson v, Jul 14 2004 *)
a[ n_] := With[{m = InverseEllipticNomeQ @ q}, SeriesCoefficient[ q Dt[ Log @ m, q], {q, 0, n}]]; (* Michael Somos, Sep 06 2012 *)
a[ n_] := (-1)^n SquaresR[ 4, n]; (* Michael Somos, Jun 12 2014 *)
a[ n_] := SeriesCoefficient[ EllipticTheta[ 4, 0, q]^4, {q, 0, n}]; (* Michael Somos, Jun 12 2014 *)
QP = QPochhammer; s = QP[q]^8/QP[q^2]^4 + O[q]^60; CoefficientList[s, q] (* Jean-François Alcover, Nov 23 2015 *)
PROG
(PARI) {a(n) = if( n<1, n==0, 8 * (-1)^n * sumdiv( n, d, if( d%4, d)))};
(PARI) {a(n) = local(A); if( n<0, 0, A = x *O (x^n); polcoeff( eta(x + A)^8 / eta(x^2 + A)^4, n))};
(Sage) A = ModularForms( Gamma0(4), 2, prec=57) . basis(); A[0] - 8*A[1]; # Michael Somos, Jun 12 2014
(Magma) A := Basis( ModularForms( Gamma0(4), 2), 57); A[1] - 8*A[2]; /* Michael Somos, Aug 21 2014 */
(Julia) # JacobiTheta4 is defined in A002448.
A096727List(len) = JacobiTheta4(len, 4)
A096727List(57) |> println # Peter Luschny, Mar 12 2018
CROSSREFS
KEYWORD
sign
AUTHOR
Michael Somos, Jul 06 2004
STATUS
approved