A215598 Expansion of phi(-x^2) * f(x)^3 in powers of x where phi(), f() are Ramanujan theta functions.

1, 3, -2, -11, 0, 10, -7, 0, 16, 6, 9, -10, -18, 0, -14, 11, 0, -22, 16, -6, 0, -3, 0, 48, 14, 0, 0, 0, -33, -26, 30, 0, 2, -16, 0, -10, -13, 0, -48, 26, 0, 0, 18, 0, 34, 19, 30, -16, 0, 0, -2, -6, 0, 22, -34, -21, 14, 42, 0, 0, -48, 0, 0, -80, 0, -22, -23, 0

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

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Formula

Expansion of q^(-1/8) * eta(q^2)^11 / (eta(q)^3 * eta(q^4)^4) in powers of q.

Euler transform of period 4 sequence [3, -8, 3, -4, ...].

a(n) = b(8*n + 1) where b() is multiplicative with b(2^e) = 0^e, b(p^e) = b(p) * b(p^(e-1)) - Kronecker(2, p) * p * b(p^(e-2)). b(8*n + 5) = 4 * i * A215596(n).

Example

G.f. = 1 + 3*x - 2*x^2 - 11*x^3 + 10*x^5 - 7*x^6 + 16*x^8 + 6*x^9 + 9*x^10 + ...
G.f. = q + 3*q^9 - 2*q^17 - 11*q^25 + 10*q^41 - 7*q^49 + 16*q^65 + 6*q^73 + 9*q^81 + ...

Prog

(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^11 / (eta(x + A)^3 * eta(x^4 + A)^4), n))};
(PARI) {a(n) = my(A, p, e, u, v, s, x, y, a0, a1); if( n<0, 0, n = n*8 + 1; A = factor(n); simplify( prod( k=1, matsize(A)[1], [p, e] = A[k, ]; if( p==2, 0, s = p * kronecker( 2, p); if( p%4==3, if( e%2, 0, (-s)^(e/2)), if( p%8==1, for( y=1, sqrtint(p\16), if( issquare( p - 16*y^2, &u), v=y; if( u%4!=1, u=-u); break)); a0 = 1; a1 = x = 2 * u * (-1)^(u\4 + v)); if( p%8==5, forstep( y=1, sqrtint(p\4), 2, if( issquare( p - 4*y^2, &v), u=y; if( u%4!=1, u=-u); if( v%4!=1, v=-v); break)); a0 = 1; a1 = x = 4 * I * u * (-1)^(v\4)); for( i=2, e, y = x*a1 - s*a0; a0=a1; a1=y); a1)))))};

Crossrefs

Cf. A215596.

Sequence in context: A074199 A153187 A248242 * A322603 A152177 A110326

Adjacent sequences: A215595 A215596 A215597 * A215599 A215600 A215601

Keyword

sign

Author

Michael Somos, Aug 16 2012

Status

approved