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A007096 Expansion of theta_3 / theta_4.
(Formerly M3332)
21
1, 4, 8, 16, 32, 56, 96, 160, 256, 404, 624, 944, 1408, 2072, 3008, 4320, 6144, 8648, 12072, 16720, 22976, 31360, 42528, 57312, 76800, 102364, 135728, 179104, 235264, 307672, 400704, 519808, 671744, 864960, 1109904, 1419456, 1809568, 2299832 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

Number of partitions of 2n into parts with 2 types c, c* of each part. The even parts appears with multiplicity 1 for each type. The odd parts appears with multiplicity 2 (cc or c*c* but not cc*, that is, no mixing is allowed). E.g., a(4)=8 because of 44*, 22*, 211, 21*1*, 2*1*1*, 2*11, 111*1*. - Noureddine Chair, Jan 27 2005

REFERENCES

J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 102.

N. J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; Eq. (34.3).

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..1000

Vaclav Kotesovec, A method of finding the asymptotics of q-series based on the convolution of generating functions, arXiv:1509.08708 [math.CO], Sep 30 2015, p. 11.

M. Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

FORMULA

Euler transform of period 4 sequence [4, -2, 4, 0, ...]. - Vladeta Jovovic, Mar 22 2005

Expansion of eta(q^2)^6 /(eta(q)^4 * eta(q^4)^2) in powers of q.

Expansion of phi(q) / phi(-q) = chi(q)^2 / chi(-q)^2 = psi(q)^2 / psi(-q)^2 = phi(-q^2)^2 / phi(-q)^2 = phi(q)^2 / phi(-q^2)^2 = chi(-q^2)^2 / chi(-q)^4 = chi(q)^4 / chi(-q^2)^2 = f(q)^2 / f(-q)^2 in powers of q where phi(), psi(), chi(), f() are Ramanujan theta functions.

G.f. A(x) satisfies 0 = f(A(x), A(x^3)) where f(u, v) = (1 - u^4) * (1 - v^4) - (1 - u*v)^4. - Michael Somos, Jan 01 2006

G.f. is a period 1 Fourier series which satisfies f(-1 / (16 t)) = (1/2) g(t) where q = exp(2 Pi i t) and g() is g.f. for A028939.

Expansion of Jacobian elliptic function 1 / sqrt(k') in powers of q. - see Fine.

G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = 1 + u^2 - 2*u*v^2. - Michael Somos, Jul 07 2005

Unique solution to f(x^2)^2 = (f(x) + 1 / f(x)) / 2 and f(0)=1, f'(0) nonzero.

G.f.: theta_3 / theta_4 = (Sum_{k} x^k^2) / (Sum_{k} (-x)^k^2) = (Product_{k>0} (1 - x^(4*k - 2)) / ((1 - x^(4*k - 1)) * (1 - x^(4*k - 3)))^2)^2.

A097243(n) = a(4*n). 8*A022577(n) = a(4*n + 2). a(n) = 4*A123655(n) if n>0. Convolution square of A080054.

Empirical: sum(exp(-Pi)^(n-1)*a(n),n=1..infinity) = 2^(1/4). - Simon Plouffe, Feb 20 2011

Empirical : sum(exp(-Pi*sqrt(2))^(n-1)*(-1)^(n+1)*a(n),n=1..infinity) = (-2+2*2^(1/2))^(1/4). - Simon Plouffe, Feb 20 2011

Empirical : sum(exp(-2*Pi)^(n-1)*a(n),n=1..infinity) = 1/2*(8+6*2^(1/2))^(1/4). - Simon Plouffe, Feb 20 2011

a(n) ~ exp(Pi*sqrt(n)) / (4*sqrt(2)*n^(3/4)). - Vaclav Kotesovec, Aug 28 2015

G.f.: exp(4*Sum_{k>=1} sigma(2*k - 1)*x^(2*k-1)/(2*k - 1)). - Ilya Gutkovskiy, Apr 19 2019

EXAMPLE

1 + 4*q + 8*q^2 + 16*q^3 + 32*q^4 + 56*q^5 + 96*q^6 + 160*q^7 + 256*q^8 + ...

MATHEMATICA

a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, q] / EllipticTheta[ 4, 0, q], {q, 0, n}] (* Michael Somos, Jul 11 2011 *)

a[ n_] := With[ {m = InverseEllipticNomeQ @ q}, SeriesCoefficient[ (1 - m)^(-1/4), {q, 0, n}]] (* Michael Somos, Jul 11 2011 *)

a[ n_] := SeriesCoefficient[( QPochhammer[ -q, q^2] / QPochhammer[ q, q^2])^2, {q, 0, n}] (* Michael Somos, Jul 11 2011 *)

a[ n_] := SeriesCoefficient[ (Product[ 1 - (-q)^k, {k, n}] / Product[ 1 - q^k, {k, n}])^2, {q, 0, n}] (* Michael Somos, Jul 11 2011 *)

nmax=60; CoefficientList[Series[Product[((1+x^(2*k+1))/(1-x^(2*k+1)))^2, {k, 0, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Aug 28 2015 *)

PROG

(PARI) {a(n) = local(A, B); if( n<0, 0, A = 1 + 4*x; for( k=2, n, B = A + x^2 * O(x^k); A += Pol(2 * subst(B, x, x^2)^2 - B - 1/B) / x / 8); polcoeff(A, n))} /* Michael Somos, Jul 07 2005*/

(PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x^2 + A)^3 / (eta(x + A)^2 * eta(x^4 + A)))^2, n))} /* Michael Somos, Jan 01 2006 */

CROSSREFS

Self-convolution of A080054. - Vladeta Jovovic, Mar 22 2005

Cf. A014969, A001936, A001938, A079006, A127391, A127392.

Cf. A022577, A080054, A097243, A123655.

Sequence in context: A131649 A003199 A189925 * A298356 A036313 A121986

Adjacent sequences:  A007093 A007094 A007095 * A007097 A007098 A007099

KEYWORD

nonn,easy

AUTHOR

N. J. A. Sloane, Simon Plouffe

STATUS

approved

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Last modified December 11 07:47 EST 2019. Contains 329914 sequences. (Running on oeis4.)