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A001936 Expansion of q^(-1/4) * (eta(q^4) / eta(q))^2 in powers of q.
(Formerly M1372 N0532)
13
1, 2, 5, 10, 18, 32, 55, 90, 144, 226, 346, 522, 777, 1138, 1648, 2362, 3348, 4704, 6554, 9056, 12425, 16932, 22922, 30848, 41282, 54946, 72768, 95914, 125842, 164402, 213901, 277204, 357904, 460448, 590330, 754368, 960948, 1220370, 1545306 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

The Cayley reference is actually to A079006. - Michael Somos, Feb 24 2011

In the mathoverflow link is the conjecture that a(n) == a(9*n + 2) (mod 4).

A000000 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

REFERENCES

A. Cayley, A memoir on the transformation of elliptic functions, Collected Mathematical Papers. Vols. 1-13, Cambridge Univ. Press, London, 1889-1897, Vol. 9, p. 128.

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

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

LINKS

T. D. Noe, Table of n, a(n) for n=0..1000

H. R. P. Ferguson, D. E. Nielsen and G. Cook, A partition formula for the integer coefficients of the theta function nome, Math. Comp., 29 (1975), 851-855.

joro, Is "OEIS A001935 Number of partitions with no even part repeated" efficiently computable mod 4?

M. Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

FORMULA

G.f.: Product ( 1 - x^k )^(-c(k)); c(k) = 2, 2, 2, 0, 2, 2, 2, 0, ....

Convolution square of A001935. A079006(n) = (-1)^n a(n).

Expansion of q^(-1/4) * (1/2) * (k / k')^(1/2) in powers of q.

Euler transform of period 4 sequence [ 2, 2, 2, 0, ...].

Given g.f. A(x), then B(q) = (q * A(q^4))^4 satisfies 0 = f(B(q), B(q^2)) where f(u, v) = (1 + 16*u) * (1 + 16*v) * v - u^2. - Michael Somos, Jul 09 2005

Given g.f. A(x), then B(q) = q * A(q^4) satisfies 0 = f(B(q), B(q^3)) where f(u, v) = (u^2 + v^2)^2 - u*v * (1 + 4*u*v)^2. - Michael Somos, Jul 09 2005

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

Equals A000009 convolved with A098613. - Gary W. Adamson, Mar 24 2011

a(9*n + 2) = a(n) + 4 * A210656(3*n). - Michael Somos, Apr 02 2012

Convolution inverse is A082304. - Michael Somos, May 16 2015

G.f. is a period 1 Fourier series which satisfies f(-1 / (64 t)) = (1/4) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A082304. - Michael Somos, May 16 2015

Expansion of f(-x^4)^2 / f(-x)^2 = psi(x^2) / phi(-x) = psi(-x)^2 / phi(-x)^2 = psi(x)^2 / phi(-x^2)^2 = psi(x^2)^2 / psi(-x)^2 = chi(x)^2 / chi(-x^2)^4 = 1 / (chi(x)^2 * chi(-x)^4) = 1 / (chi(-x)^2 * chi(-x^2)^2) in powers of q where phi(), psi(), chi(), f() are Ramanujan theta functions. - Michael Somos, May 16 2015

EXAMPLE

G.f. = 1 + 2*x + 5*x^2 + 10*x^3 + 18*x^4 + 32*x^5 + 55*x^6 + 90*x^7 + 144*x^8 + ...

G.f. = q + 2*q^5 + 5*q^9 + 10*q^13 + 18*q^17 + 32*q^21 + 55*q^25 + 90*q^29 + ...

MAPLE

with(numtheory): etr:= proc(p) local b; b:=proc(n) option remember; local d, j; if n=0 then 1 else add(add(d*p(d), d=divisors(j)) *b(n-j), j=1..n)/n fi end end: a:= etr(n-> [2, 2, 2, 0] [modp(n-1, 4)+1]): seq(a(n), n=0..40); # Alois P. Heinz, Sep 08 2008

MATHEMATICA

m = 38; CoefficientList[ Series[ Product[ (1 - x^(4*k))/(1 - x^k), {k, 1, m}]^2 , {x, 0, m}], x] (* Jean-Fran├žois Alcover, Sep 02 2011, after g.f. *)

a[ n_] := SeriesCoefficient[ (EllipticTheta[ 2, 0, x] / EllipticTheta[ 4, 0, x]) / (2 x^(1/4)), {x, 0, n}]; (* Michael Somos, May 16 2015 *)

a[ n_] := SeriesCoefficient[ (Product[ 1 - x^k, {k, 4, n, 4}] / Product[ 1 - x^k, {k, n}])^2, {x, 0, n}]; (* Michael Somos, May 16 2015 *)

a[ n_] := SeriesCoefficient[ (QPochhammer[ x^4] / QPochhammer[ x])^2, {x, 0, n}]; (* Michael Somos, May 16 2015 *)

a[ n_] := SeriesCoefficient[ (QPochhammer[ -x, x] QPochhammer[ -x^2, x^2])^2, {x, 0, n}]; (* Michael Somos, May 16 2015 *)

PROG

(PARI) {a(n) = if( n<0, 0, polcoeff( (eta(x^4 + x * O(x^n)) / eta(x + x * O(x^n)))^2, n))};

(PARI) {a(n) = if( n<0, 0, polcoeff( prod(k=1, n, 1 / if(k%4, 1 - x^k, 1), 1 + x * O(x^n))^2, n))};

CROSSREFS

Cf. A001935, A079006, A0082304, A098613, A127391, A127392, A210656.

Sequence in context: A185721 A103577 A079006 * A246712 A224364 A246840

Adjacent sequences:  A001933 A001934 A001935 * A001937 A001938 A001939

KEYWORD

nonn,changed

AUTHOR

N. J. A. Sloane

STATUS

approved

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Last modified May 25 07:27 EDT 2015. Contains 257806 sequences.