A259285 Expansion of psi(x^2) * f(x, x^7) in powers of x where psi(), f(,) are Ramanujan theta functions.

1, 1, 1, 1, 0, 0, 1, 2, 0, 1, 1, 0, 2, 2, 0, 0, 1, 0, 0, 1, 1, 1, 2, 0, 1, 0, 0, 2, 1, 1, 2, 1, 0, 1, 1, 0, 0, 1, 0, 1, 1, 0, 2, 1, 0, 1, 0, 2, 0, 1, 0, 1, 3, 0, 1, 0, 1, 3, 1, 0, 0, 0, 0, 1, 2, 1, 1, 0, 0, 1, 0, 0, 2, 1, 0, 1, 1, 0, 2, 1, 0, 0, 3, 1, 0, 1, 0

Offset: 0

Author

Michael Somos, Jun 23 2015

Keywords

• nonn

Comments

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

Examples

			G.f. = 1 + x + x^2 + x^3 + x^6 + 2*x^7 + x^9 + x^10 + 2*x^12 + 2*x^13 + ...
G.f. = q^13 + q^29 + q^45 + q^61 + q^109 + 2*q^125 + q^157 + q^173 + ...
		

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

Crossrefs

Cf. A000161, A025426, A025435, A025441, A134343, A259287.

Programs

Mathematica

a[ n_] := SeriesCoefficient[ QPochhammer[ -x^1, x^8] QPochhammer[ -x^2, x^8] QPochhammer[ -x^6, x^8] QPochhammer[ -x^7, x^8] QPochhammer[x^8]^2, {x, 0, n}];
a[ n_] := SeriesCoefficient[ Product[ (1 + x^(8 k - 1)) (1 + x^(8 k - 2)) (1 + x^(8 k - 6)) (1 + x^(8 k - 7)) (1 - x^(8 k))^2, {k, Ceiling[n/8]}], {x, 0, n}];

PARI

{a(n) = my(m, s, x, c); if( n<0, 0, s = sqrtint(m = 16*n + 13); for(u = (s+3)\-8, (s-3)\8, if( issquare( m - (8*u + 3)^2, &x) && (x%8==2 || x%8==6), c++))); c};

PARI

{a(n) = if( n<0, 0, polcoeff( prod(k=1, n, (1 - x^k)^[ 2, -1, 0, 0, 1, 0, -1, -1, 2, -1, -1, 0, 1, 0, 0, -1][k%16 + 1], 1 + x * O(x^n)), n))};

Formula

Number of solutions to 16*n + 13 = (8*u + 3)^2 + (8*v + 2)^2 where u,v in Z.

Euler transform of period 16 sequence [ 1, 0, 0, -1, 0, 1, 1, -2, 1, 1, 0, -1, 0, 0, 1, -2, ...].

a(9*n + 2) = A259287(n). a(9*n + 5) = a(9*n + 8) = 0.

-2 * a(n) = A134343(4*n + 3). a(n) = A000161(16*n + 13) = A025426(16*n + 13) = A025435(16*n + 13) = A025441(16*n + 13).