A092833 Expansion of q / (chi(-q) * chi(-q^23)) in powers of q where chi() is a Ramanujan theta function.

1, 1, 1, 2, 2, 3, 4, 5, 6, 8, 10, 12, 15, 18, 22, 27, 32, 38, 46, 54, 64, 76, 89, 105, 123, 143, 167, 194, 225, 260, 301, 346, 398, 458, 524, 600, 686, 782, 891, 1014, 1151, 1306, 1480, 1674, 1892, 2137, 2409, 2713, 3053, 3431, 3852, 4322, 4842, 5421, 6064, 6776

Offset

1,4

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 = 1..1000

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Formula

G.f.: x * (Product_{k>0} (1 + x^k) * (1 + x^(23*k))).

Expansion of eta(q^2) * eta(q^46) / (eta(q) * eta(q^23)) in powers of q.

Euler transform of period 46 sequence with g.f. x / (1 - x^2) + x^23 / (1 - x^46).

G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = u^2 - v - 2 * u*v * (1 + v).

G.f. is a period 1 Fourier series which satisfies f(-1 / (46 t)) = (1/2) g(t) where q = exp(2 Pi i t) and g(t) is the g.f. for A132322.

Convolution inverse of A132322.

a(n) = A112216(2*n). - Michael Somos, Aug 11 2015

a(n) ~ exp(2*Pi*sqrt(2*n/23)) / (2^(7/4) * 23^(1/4) * n^(3/4)). - Vaclav Kotesovec, Sep 07 2015

Example

G.f. = q + q^2 + q^3 + 2*q^4 + 2*q^5 + 3*q^6 + 4*q^7 + 5*q^8 + 6*q^9 + 8*q^10 + ...

Mathematica

a[n_] := Coefficient[ Series[ x*Product[(1 + x^k)*(1 + x^(23*k)), {k, 1, n}], {x, 0, n}], x, n]; Table[a[n], {n, 1, 56}] (* Jean-François Alcover, Jan 28 2013, from 1st formula *)
a[ n_] := SeriesCoefficient[ q Product[ (1 + q^k) (1 + q^(23 k)), {k, n}], {q, 0, n}]; (* Michael Somos, Aug 11 2015 *)
a[ n_] := SeriesCoefficient[ q (QPochhammer[ -q, q]  QPochhammer[ -q^23, q^23]), {q, 0, n}]; (* Michael Somos, Aug 11 2015 *)

Prog

(PARI) {a(n) = my(A, m); if( n<0, 0, A = x + O(x^2); m=1; while( m<=n, m*=2; A = subst(A, x, x^2); A = A + A^2 + sqrt(A + (A + A^2)^2)); polcoeff(A, n))};
(PARI) {a(n) = my(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( eta(x^2 + A) * eta(x^46 + A) / eta(x + A) / eta(x^23 + A), n))};

Crossrefs

Cf. A112216, A132322.

Sequence in context: A000009 A081360 A117409 * A280664 A100926 A351008

Adjacent sequences: A092830 A092831 A092832 * A092834 A092835 A092836

Keyword

nonn

Author

Michael Somos, Mar 06 2004

Status

approved