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A260308 Expansion of psi(x) * phi(x^3) in powers of x where phi(), psi() are Ramanujan theta functions. 6
1, 1, 0, 3, 2, 0, 3, 0, 0, 2, 1, 0, 2, 4, 0, 3, 0, 0, 4, 0, 0, 1, 2, 0, 2, 0, 0, 4, 3, 0, 2, 2, 0, 4, 0, 0, 1, 2, 0, 2, 2, 0, 2, 0, 0, 1, 0, 0, 8, 2, 0, 2, 0, 0, 2, 3, 0, 2, 4, 0, 0, 0, 0, 4, 0, 0, 1, 2, 0, 4, 0, 0, 2, 0, 0, 2, 4, 0, 5, 0, 0, 4, 2, 0, 2, 2, 0 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,4

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 q^(-1/8) * eta(q^2)^2 * eta(q^6)^5 / (eta(q) * eta(q^3)^2 * eta(q^12)^2) in powers of q.

Euler transform of period 12 sequence [ 1, -1, 3, -1, 1, -4, 1, -1, 3, -1, 1, -2, ...].

a(n) = A259668(2*n) = A128580(4*n) = A129402(4*n) = A134177(4*n) = A190615(4*n) = A115660(8*n + 1) = A128581(8*n + 1) = A192013(8*n + 1).

EXAMPLE

G.f. = 1 + x + 3*x^3 + 2*x^4 + 3*x^6 + 2*x^9 + x^10 + 2*x^12 + 4*x^13 + ...

G.f. = q + q^9 + 3*q^25 + 2*q^33 + 3*q^49 + 2*q^73 + q^81 + 2*q^97 + ...

MATHEMATICA

a[ n_] := If[ n < 0, 0, DivisorSum[ 8 n + 1, KroneckerSymbol[ -6, #] &]];

a[ n_] := If[ n < 0, 0, Times @@ (Which[ # <= 3, Mod[#, 2], Mod[#, 24] > 12, 1 - Mod[#2, 2], True, (#2  + 1) KroneckerSymbol[ 3, #]^#2] & @@@ FactorInteger @ (8 n + 1))];

a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, x^3] EllipticTheta[ 2, 0, x^(1/2)] / (2 x^(1/8)), {x, 0, n}];

PROG

(PARI) {a(n) = if( n<0, 0, sumdiv( 8*n + 1, d, kronecker( -6, d)))};

(PARI) {a(n) = my(A, p, e); if( n<0, 0, factor(8*n + 1); prod( k=1, matsize(A)[1], [p, e] = A[k, ]; if( p==2, 0, p==3, 1, p%24>12, !(e%2), (e+1) * kronecker(3, p)^e)))};

(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^2 * eta(x^6 + A)^5 / (eta(x + A) * eta(x^3 + A)^2 * eta(x^12 + A)^2), n))};

CROSSREFS

Cf. A115660, A128580, A128581, A129402, A134177, A190615, A192013, A259668.

Sequence in context: A075115 A273528 A085080 * A079714 A286889 A286368

Adjacent sequences:  A260305 A260306 A260307 * A260309 A260310 A260311

KEYWORD

nonn

AUTHOR

Michael Somos, Jul 22 2015

STATUS

approved

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Last modified February 22 20:16 EST 2019. Contains 320404 sequences. (Running on oeis4.)