A073252 Coefficients of replicable function number "48g".

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

Author

Michael Somos, Jul 22 2002

Keywords

• nonn
• easy

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.

Examples

			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 + ...
		

References

• T. J. I'a. Bromwich, Introduction to the Theory of Infinite Series, Macmillan, 2nd. ed. 1949, p. 116, q_2^2.

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..1000
• D. Foata and G.-N. Han, Jacobi and Watson Identities Combinatorially Revisited
• D. Ford, J. McKay and S. P. Norton, More on replicable functions, Commun. Algebra 22, No. 13, 5175-5193 (1994).
• Michael Somos, Introduction to Ramanujan theta functions
• Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
• Index entries for McKay-Thompson series for Monster simple group

Crossrefs

Cf. A000122, A000700, A010054, A022597, A121373, A226622, A226635.

Programs

Magma

m:=80;
R:=PowerSeriesRing(Integers(), m);
Coefficients(R!( ( (&*[1 + x^(2*j+1): j in [0..m+2]]) )^2 )); // G. C. Greubel, Sep 07 2023

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 *)

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))};

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. = 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

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

Extensions

Comments from Len Smiley.

New name from Michael Somos, Nov 03 2019