A187154 Expansion of psi(x^4) / phi(-x) in powers of x where phi(), psi() are Ramanujan theta functions.

1, 2, 4, 8, 15, 26, 44, 72, 114, 178, 272, 408, 605, 884, 1276, 1824, 2580, 3616, 5028, 6936, 9498, 12922, 17468, 23472, 31369, 41700, 55156, 72616, 95172, 124202, 161436, 209016, 269616, 346562, 443952, 566856, 721530, 915642, 1158608, 1461968, 1839789

Offset: 0

Author

Michael Somos, Mar 08 2011

Keywords

• nonn
• easy

Comments

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

Since phi(-x) = 1 + 2*Sum_{k >= 1} (-1)^k*x^(k^2) == 1 (mod 2), it follows that the g.f. psi(x^4) / phi(-x) == psi(x^4) == Sum_{k >= 0} x^(2*k*(k+1)) (mod 2). Hence a(n) is odd iff n = 2*k*(k + 1) for some nonnegative integer k. - Peter Bala, Jan 07 2025

Examples

			1 + 2*x + 4*x^2 + 8*x^3 + 15*x^4 + 26*x^5 + 44*x^6 + 72*x^7 + 114*x^8 + ...
q + 2*q^3 + 4*q^5 + 8*q^7 + 15*q^9 + 26*q^11 + 44*q^13 + 72*q^15 + 114*q^17 + ...
		

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. A001935, A001936, A022568, A093085, A107035.

Programs

Mathematica

nmax = 50; CoefficientList[Series[Product[(1 + x^k)^2 * (1 + x^(2*k)) * (1 + x^(4*k))^2, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Sep 10 2015 *)
a[n_]:= SeriesCoefficient[EllipticTheta[2, 0, q^2]/(2*Sqrt[q]* EllipticTheta[3, 0, -q]), {q, 0, n}]; Table[A187154[n], {n, 0, 50}] (* G. C. Greubel, Dec 04 2017 *)

PARI

{a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A) * eta(x^8 + A)^2 / (eta(x + A)^2 * eta(x^4 + A)), n))}

Formula

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

Euler transform of period 8 sequence [ 2, 1, 2, 2, 2, 1, 2, 0, ...].

G.f. is a period 1 Fourier series which satisfies f(-1 / (32 t)) = 32^(-1/2) g(t) where q = exp(2 Pi i t) and g() is g.f. for A093085.

Convolution inverse of A093085. Convolution square is A107035.

a(n) ~ exp(sqrt(n)*Pi)/(16*n^(3/4)). - Vaclav Kotesovec, Sep 10 2015