A073252 - OEIS

%I #59 Feb 16 2025 08:32:46

%S 1,2,1,2,4,4,5,6,9,12,13,16,21,26,29,36,46,54,62,74,90,106,122,142,

%T 171,200,227,264,311,358,408,470,545,626,709,810,933,1062,1198,1362,

%U 1555,1760,1980,2238,2536,2858,3205,3602,4063,4560,5092,5704,6400,7150,7966

%N Coefficients of replicable function number "48g".

%C Old name was: McKay-Thompson series of class 48g for the Monster group.

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

%C Combinatorial interpretation of sequence: [ X1, X2 ] = 2 strictly increasing sequences (possibly null) of odd positive integers; a(n) = #pairs with sum of entries = n.

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

%H Seiichi Manyama, <a href="/A073252/b073252.txt">Table of n, a(n) for n = 0..1000</a>

%H D. Foata and G.-N. Han, <a href="https://irma.math.unistra.fr/~foata/paper/pub81a.pdf">Jacobi and Watson Identities Combinatorially Revisited</a>

%H D. Ford, J. McKay and S. P. Norton, <a href="http://dx.doi.org/10.1080/00927879408825127">More on replicable functions</a>, Commun. Algebra 22, No. 13, 5175-5193 (1994).

%H Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>

%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>

%H <a href="/index/Mat#McKay_Thompson">Index entries for McKay-Thompson series for Monster simple group</a>

%F G.f.: 1 / (Prod_{k>0} 1 + (-x)^k)^2 = (Prod_{k>0} 1 + x^(2*k - 1))^2.

%F Expansion of q^(1/12) * (eta(q^2)^2 / (eta(q) * eta(q^4)))^2 in powers of q.

%F 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.

%F Euler transform of period 4 sequence [2, -2, 2, 0, ...].

%F Equals the convolution square of A000700.

%F a(n) = (-1)^n * A022597(n).

%F a(n) ~ exp(Pi*sqrt(n/3)) / (2^(3/2) * 3^(1/4) * n^(3/4)). - _Vaclav Kotesovec_, Aug 27 2015

%F G.f.: exp(2*Sum_{k>=1} x^k/(k*(1 - (-x)^k))). - _Ilya Gutkovskiy_, Jun 07 2018

%F a(2*n) = A226622(n). a(2*n + 1) = 2 * A226635(n). - _Michael Somos_, Nov 03 2019

%e a(4) = 4: [ (1),(3) ],[ (3),(1) ],[ (),(1,3) ],[ (1,3),() ]

%e 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 + ...

%e G.f. = 1/q + 2*q^11 + q^23 + 2*q^35 + 4*q^47 + 4*q^59 + 5*q^71 + 6*q^83 + ...

%t nmax = 50; CoefficientList[Series[Product[(1 + x^(2*k+1))^2, {k, 0, nmax}], {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Aug 27 2015 *)

%t 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 *)

%t a[ n_] := SeriesCoefficient[ QPochhammer[ -x, x^2]^2, {x, 0, n}]; (* _Michael Somos_, Nov 03 2019 *)

%o (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))};

%o (PARI) {a(n) = if( n<0, 0, polcoeff( 1 / prod( i=1, n, 1 + (-x)^i, 1 + x * O(x^n))^2, n))};

%o (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))};

%o (Magma)

%o m:=80;

%o R<x>:=PowerSeriesRing(Integers(), m);

%o Coefficients(R!( ( (&*[1 + x^(2*j+1): j in [0..m+2]]) )^2 )); // _G. C. Greubel_, Sep 07 2023

%o (SageMath)

%o from sage.modular.etaproducts import qexp_eta

%o m=80

%o def f(x): return qexp_eta(QQ[['q']], m+2).subs(q=x)

%o def A073252_list(prec):

%o     P.<x> = PowerSeriesRing(QQ, prec)

%o     return P( (f(x^2)^2/(f(x)*f(x^4)))^2 ).list()

%o A073252_list(m) # _G. C. Greubel_, Sep 07 2023

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

%K nonn,easy

%O 0,2

%A _Michael Somos_, Jul 22 2002

%E Comments from _Len Smiley_.

%E New name from _Michael Somos_, Nov 03 2019