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A204010 Expansion of f(-x^12) * phi(-x) in powers of x where f(), phi() are Ramanujan theta functions. 2
1, -2, 0, 0, 2, 0, 0, 0, 0, -2, 0, 0, -1, 2, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, -1, 0, 0, 0, -4, 0, 0, 0, 0, 2, 0, 0, 2, 2, 0, 0, -2, 0, 0, 0, 0, 0, 0, 0, -2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 4, 0, 0, 0, 0, -2, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, -2, 0 (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..2500

M. Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

FORMULA

Expansion of q^(-1/2) * eta(q)^2 * eta(q^12) / eta(q^2) in powers of q.

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

G.f.: (Product_{k>0} 1 - x^(12*k)) * (Sum_{k in Z} (-x)^k^2).

a(3*n + 2) = a(4*n + 2) = a(4*n + 3) = 0.

EXAMPLE

G.f. = 1 - 2*x + 2*x^4 - 2*x^9 - x^12 + 2*x^13 + 2*x^21 - x^24 - 4*x^28 + 2*x^33 + ...

G.f. = q - 2*q^3 + 2*q^9 - 2*q^19 - q^25 + 2*q^27 + 2*q^43 - q^49 - 4*q^57 + 2*q^67 + ...

MATHEMATICA

a[ n_] := SeriesCoefficient[ QPochhammer[ q^12] EllipticTheta[ 4, 0, q], {q, 0, n}];

a[ n_] := Module[ {m = 2 n + 1, i}, If[ n < 0, 0, Sum[i = Sqrt[ m - 2 j^2]; If[ IntegerQ @ i, (1 + Boole[ j > 0]) (-1)^j JacobiSymbol[ 12, i], 0], {j, 0, Sqrt[ m / 2]}]]];

PROG

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

(PARI) {a(n) = my(m, i); if( n<0, 0, m = 2*n + 1; sum( j=0, sqrtint( m \ 2), if( issquare( m - 2 * j^2, &i), (1 + (j>0)) * (-1)^j * kronecker( 12, i), 0)))};

(PARI) {a(n) = my(A, p, e, i); if( n<0, 0, n = 2*n + 1; A = factor(n); [1, 2*I, 0, I, 2, 0][ n\2%6 + 1] * prod( k=1, matsize(A)[1], [p, e] = A[k, ]; if( p==2, 0, p==3, -(-I)^e, p%24 == 1 || p%24 == 19, forstep( j = 3*(p%24>1), sqrtint( p \ 2), 6, if( issquare( p - 2*j^2, &i), break)); kronecker( 12, i)^e * if(p%24>1, I^e, 1) * (e+1), if( e%2, 0, I^((p%24<12)*e)))))};

CROSSREFS

Sequence in context: A255258 A260671 A033725 * A033723 A138811 A107494

Adjacent sequences:  A204007 A204008 A204009 * A204011 A204012 A204013

KEYWORD

sign

AUTHOR

Michael Somos, Jun 03 2013

STATUS

approved

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Last modified April 23 02:15 EDT 2019. Contains 322380 sequences. (Running on oeis4.)