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A122856 Expansion of f(x, x^5)^2 in powers of x where f(, ) is Ramanujan's general theta function. 33
1, 2, 1, 0, 0, 2, 2, 0, 2, 2, 1, 0, 0, 2, 0, 0, 3, 2, 0, 0, 0, 4, 2, 0, 2, 0, 2, 0, 0, 2, 0, 0, 1, 2, 2, 0, 0, 2, 2, 0, 2, 4, 1, 0, 0, 2, 0, 0, 2, 2, 0, 0, 0, 0, 2, 0, 4, 2, 0, 0, 0, 4, 0, 0, 2, 2, 3, 0, 0, 0, 2, 0, 2, 4, 0, 0, 0, 2, 0, 0, 1, 2, 0, 0, 0, 2, 4, 0, 0, 2, 2, 0, 0, 2, 0, 0, 4, 2, 2, 0, 0, 4, 0, 0, 2 (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).

LINKS

G. C. Greubel, Table of n, a(n) for n = 0..1000

M. Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

FORMULA

Expansion of (chi(x) * psi(-x^3))^2 in powers of x where chi(), psi() are Ramanujan theta functions.

Expansion of q^(-2/3) * (eta(q^2)^2 * eta(q^3) * eta(q^12) / (eta(q) * eta(q^4) * eta(q^6)))^2 in powers of q.

Euler transform of period 12 sequence [2, -2, 0, 0, 2, -2, 2, 0, 0, -2, 2, -2, ...].

a(4*n + 3) = a(8*n + 4) = 0. a(n) = A002654(3*n + 2) = A035154(3*n + 2) = A113446(3*n + 2). a(2*n) = A122865(n). a(4*n + 1) = 2 * A121444(n). a(4*n + 2) = A122856(n).

a(n) = (-1)^n * A258278(n). Convolution square of A089801.

EXAMPLE

G.f. = 1 + 2*x + x^2 + 2*x^5 + 2*x^6 + 2*x^8 + 2*x^9 + x^10 + 2*x^13 + ...

G.f. = q^2 + 2*q^5 + q^8 + 2*q^17 + 2*q^20 + 2*q^26 + 2*q^29 + q^32 + ...

MATHEMATICA

a[ n_] := If[ n < 0, 0, With[ {m = 3 n + 2}, Sum[ KroneckerSymbol[ -4, d], {d, Divisors@m}]]]; (* Michael Somos, Nov 14 2011 *)

QP = QPochhammer; s = (QP[q^2]^2*QP[q^3]*(QP[q^12]/(QP[q]*QP[q^4]*QP[q^6]) ))^2 + O[q]^105; CoefficientList[s, q] (* Jean-François Alcover, Nov 30 2015, adapted from PARI *)

a[ n_] := SeriesCoefficient[ (EllipticTheta[ 3, 0, x^(1/3)] - EllipticTheta[ 3, 0, x^3])^2 / (4 x^(2/3)), {x, 0, n}]; (* Michael Somos, Jan 19 2017 *)

a[ n_] := SeriesCoefficient[ (QPochhammer[ -x, x^2] EllipticTheta[ 2, Pi/4, x^(3/2)])^2 / (2 x^(3/4)), {x, 0, n}]; (* Michael Somos, Jan 19 2017 *)

PROG

(PARI) {a(n) = if( n<0, 0, n = 3*n+2; sumdiv(n, d, (d%4==1) - (d%4==3)))};

(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x^2 + A)^2 * eta(x^3 + A) * eta(x^12 + A) / (eta(x + A) * eta(x^4 + A) * eta(x^6 + A)))^2, n))};

CROSSREFS

Cf. A002654, A035154, A089801, A113446, A121444, A122865, A258278.

Sequence in context: A008626 A058626 A258278 * A055791 A245842 A300574

Adjacent sequences:  A122853 A122854 A122855 * A122857 A122858 A122859

KEYWORD

nonn

AUTHOR

Michael Somos, Sep 14 2006

STATUS

approved

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Last modified April 19 16:58 EDT 2019. Contains 322283 sequences. (Running on oeis4.)