A213624 Expansion of psi(x)^2 * psi(x^4) in powers of x where psi() is a Ramanujan theta function.

1, 2, 1, 2, 3, 2, 4, 4, 2, 2, 5, 4, 2, 6, 3, 6, 7, 2, 5, 4, 5, 6, 6, 2, 5, 10, 3, 6, 10, 4, 6, 8, 3, 8, 7, 6, 7, 6, 4, 6, 11, 6, 9, 10, 3, 6, 14, 4, 8, 10, 8, 10, 5, 6, 4, 16, 7, 4, 10, 4, 13, 14, 7, 8, 8, 6, 10, 12, 7, 12, 15, 8, 8, 10, 4, 6, 17, 6, 10, 10

Offset

0,2

Comments

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

Dickson writes that Liouville proved several related results about sums of triangular number. In particular, that every nonnegative integer is the sum of t1 + t2 + 4*t3 where t1, t2, t3 are trianglular numbers. - Michael Somos, Feb 10 2020

References

L. E. Dickson, History of the Theory of Numbers. Carnegie Institute Public. 256, Washington, DC, Vol. 1, 1919; Vol. 2, 1920; Vol. 3, 1923, see vol. 2, p. 23.

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

Euler transform of period 8 sequence [2, -2, 2, -1, 2, -2, 2, -3, ...].

Empirical: Sum_{n>=0} a(n) / exp(n*Pi) = (1/32) * exp(3*Pi/4) * sqrt(2) * Gamma(5/8)^3 * (1+sqrt(2)) / Pi^(3/4) / Gamma(7/8)^3 = A388763. - Simon Plouffe, Sep 18 2025

Example

G.f. = 1 + 2*x + x^2 + 2*x^3 + 3*x^4 + 2*x^5 + 4*x^6 + 4*x^7 + 2*x^8 + 2*x^9 + ...
G.f. = q^3 + 2*q^7 + q^11 + 2*q^15 + 3*q^19 + 2*q^23 + 4*q^27 + 4*q^31 + 2*q^35 + ...

Mathematica

a[ n_] := SeriesCoefficient[ 1/8 EllipticTheta[ 2, 0, q]^2 EllipticTheta[ 2, 0, q^4], {q, 0, 2 n + 3/2}];

Prog

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

Crossrefs

Cf. A008443, A045818,

Sequence in context: A029826 A192185 A246833 * A080845 A290370 A029166

Adjacent sequences: A213621 A213622 A213623 * A213625 A213626 A213627

Keyword

nonn

Author

Michael Somos, Jun 16 2012

Status

approved