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A073252
Coefficients of replicable function number "48g".
9
1, 2, 1, 2, 4, 4, 5, 6, 9, 12, 13, 16, 21, 26, 29, 36, 46, 54, 62, 74, 90, 106, 122, 142, 171, 200, 227, 264, 311, 358, 408, 470, 545, 626, 709, 810, 933, 1062, 1198, 1362, 1555, 1760, 1980, 2238, 2536, 2858, 3205, 3602, 4063, 4560, 5092, 5704, 6400, 7150, 7966
OFFSET
0,2
COMMENTS
Old name was: McKay-Thompson series of class 48g for the Monster group.
Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
Combinatorial interpretation of sequence: [ X1, X2 ] = 2 strictly increasing sequences (possibly null) of odd positive integers; a(n) = #pairs with sum of entries = n.
REFERENCES
T. J. I'a. Bromwich, Introduction to the Theory of Infinite Series, Macmillan, 2nd. ed. 1949, p. 116, q_2^2.
LINKS
D. Ford, J. McKay and S. P. Norton, More on replicable functions, Commun. Algebra 22, No. 13, 5175-5193 (1994).
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
FORMULA
G.f.: 1 / (Prod_{k>0} 1 + (-x)^k)^2 = (Prod_{k>0} 1 + x^(2*k - 1))^2.
Expansion of q^(1/12) * (eta(q^2)^2 / (eta(q) * eta(q^4)))^2 in powers of q.
Expansion of chi(q)^2 = phi(q) / f(-q^2) = f(q) / psi(-q) = (phi(q) / f(q))^2 = (psi(q) / f(-q^4))^2 = (f(-q^2) / psi(-q))^2 = (phi(-q^2) / f(-q))^2 = (f(q) / f(-q^2))^2 in powers of q where phi(), psi(), chi(), f() are Ramanujan theta functions.
Euler transform of period 4 sequence [2, -2, 2, 0, ...].
Equals the convolution square of A000700.
a(n) = (-1)^n * A022597(n).
a(n) ~ exp(Pi*sqrt(n/3)) / (2^(3/2) * 3^(1/4) * n^(3/4)). - Vaclav Kotesovec, Aug 27 2015
G.f.: exp(2*Sum_{k>=1} x^k/(k*(1 - (-x)^k))). - Ilya Gutkovskiy, Jun 07 2018
a(2*n) = A226622(n). a(2*n + 1) = 2 * A226635(n). - Michael Somos, Nov 03 2019
Empirical: Sum_{n>=0} a(n) / exp(n*Pi) = exp(-Pi/12) * sqrt(2) = A388453. - Simon Plouffe, Sep 17 2025
EXAMPLE
a(4) = 4: [ (1),(3) ],[ (3),(1) ],[ (),(1,3) ],[ (1,3),() ]
G.f. = 1 + 2*x + x^2 + 2*x^3 + 4*x^4 + 4*x^5 + 5*x^6 + 6*x^7 + 9*x^8 + 12*x^9 + ...
G.f. = 1/q + 2*q^11 + q^23 + 2*q^35 + 4*q^47 + 4*q^59 + 5*q^71 + 6*q^83 + ...
MATHEMATICA
nmax = 50; CoefficientList[Series[Product[(1 + x^(2*k+1))^2, {k, 0, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Aug 27 2015 *)
QP = QPochhammer; s = (QP[q^2]^2 / (QP[q] * QP[q^4]))^2 + O[q]^60; CoefficientList[s, q] (* Jean-François Alcover, Nov 14 2015, adapted from PARI *)
a[ n_] := SeriesCoefficient[ QPochhammer[ -x, x^2]^2, {x, 0, n}]; (* Michael Somos, Nov 03 2019 *)
PROG
(PARI) {a(n) = if( n<0, 0, polcoeff( prod( i=1, (1+n)\2, 1 + x^(2*i - 1), 1 + x * O(x^n))^2, n))};
(PARI) {a(n) = if( n<0, 0, polcoeff( 1 / prod( i=1, n, 1 + (-x)^i, 1 + x * O(x^n))^2, n))};
(PARI) {a(n) = my(A); if( n<0, 0, A = x*O(x^n); polcoeff( (eta(x^2 + A)^2 / eta(x + A) / eta(x^4 + A))^2, n))};
(Magma)
m:=80;
R<x>:=PowerSeriesRing(Integers(), m);
Coefficients(R!( ( (&*[1 + x^(2*j+1): j in [0..m+2]]) )^2 )); // G. C. Greubel, Sep 07 2023
(SageMath)
from sage.modular.etaproducts import qexp_eta
m=80
def f(x): return qexp_eta(QQ[['q']], m+2).subs(q=x)
def A073252_list(prec):
P.<x> = PowerSeriesRing(QQ, prec)
return P( (f(x^2)^2/(f(x)*f(x^4)))^2 ).list()
A073252_list(m) # G. C. Greubel, Sep 07 2023
KEYWORD
nonn,easy
AUTHOR
Michael Somos, Jul 22 2002
EXTENSIONS
Comments from Len Smiley.
New name from Michael Somos, Nov 03 2019
STATUS
approved