A022599 - OEIS

%I #40 Mar 12 2021 22:24:41

%S 1,-4,6,-8,17,-28,38,-56,84,-124,172,-232,325,-448,594,-784,1049,

%T -1388,1796,-2320,3005,-3864,4912,-6216,7877,-9940,12430,-15488,19309,

%U -23972,29580,-36408,44766,-54876,66978,-81536,99150,-120272,145374,-175344,211242

%N Expansion of Product_{m>=1} (1+q^m)^(-4).

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

%C McKay-Thompson series of class 12J for the Monster group.

%D T. J. I'a. Bromwich, Introduction to the Theory of Infinite Series, Macmillan, 2nd. ed. 1949, p. 116, q_2^4.

%H Seiichi Manyama, <a href="/A022599/b022599.txt">Table of n, a(n) for n = 0..10000</a> (terms 0..1000 from Alois P. Heinz)

%H J. H. Conway and S. P. Norton, <a href="http://blms.oxfordjournals.org/content/11/3/308.extract">Monstrous Moonshine</a>, Bull. Lond. Math. Soc. 11 (1979) 308-339.

%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 Vaclav Kotesovec, <a href="http://arxiv.org/abs/1509.08708">A method of finding the asymptotics of q-series based on the convolution of generating functions</a>, arXiv:1509.08708 [math.CO], Sep 30 2015, p. 13.

%H J. McKay and H. Strauss, <a href="http://dx.doi.org/10.1080/00927879008823911">The q-series of monstrous moonshine and the decomposition of the head characters</a>, Comm. Algebra 18 (1990), no. 1, 253-278.

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

%H Eric Weisstein's World of Mathematics, <a href="http://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 Expansion of chi(-x)^4 in powers of x where chi() is a Ramanujan theta function.

%F Expansion of q^(1/6) * (eta(q) / eta(q^2))^4 in powers of q.

%F Euler transform of period 2 sequence [ -4, 0, ...]. - _Michael Somos_, Apr 26 2008

%F Given G.f. A(x) then B(q) = (A(q^6) / q)^2 satisfies 0 = f(B(q), B(q^2)) where f(u, v) = u * (16 + u * v) - v^2. - _Michael Somos_, Apr 26 2008

%F Given G.f. A(x) then B(q) = A(q^6) / q satisfies 0 = f(B(q), B(q^2), B(q^4)) where f(u, v, w) = 4 * v * (v + u^2) - w^2 * (v - u^2). - _Michael Somos_, Apr 26 2008

%F G.f. is a period 1 Fourier series which satisfies f(-1 / (72 t)) = 4 g(t) where q = exp(2 Pi i t) and g() is the g.f. of A022569.

%F Convolution inverse is A022569. Convolution square of A022597. Convolution square is A007259.

%F a(n) ~ (-1)^n * exp(Pi*sqrt(2*n/3)) / (2 * 6^(1/4) * n^(3/4)). - _Vaclav Kotesovec_, Aug 27 2015

%F a(0) = 1, a(n) = -(4/n)*Sum_{k=1..n} A000593(k)*a(n-k) for n > 0. - _Seiichi Manyama_, Apr 03 2017

%F G.f.: exp(-4*Sum_{k>=1} (-1)^(k+1)*x^k/(k*(1 - x^k))). - _Ilya Gutkovskiy_, Feb 06 2018

%e G.f. = 1 - 4*x + 6*x^2 - 8*x^3 + 17*x^4 - 28*x^5 + 38*x^6 - 56*x^7 + ...

%e T12J = 1/q - 4*q^5 + 6*q^11 - 8*q^17 + 17*q^23 - 28*q^29 + 38*q^35 + ...

%p with(numtheory):

%p a:= proc(n) option remember; `if`(n=0, 1, add(add(

%p       -4*irem(d, 2)*d, d=divisors(j))*a(n-j), j=1..n)/n)

%p     end:

%p seq(a(n), n=0..50);  # _Alois P. Heinz_, May 02 2014

%t a[ n_] := SeriesCoefficient[ (QPochhammer[ x] / QPochhammer[x^2])^4, {x, 0, n}]; (* _Michael Somos_, Jul 05 2014 *)

%t nmax = 50; CoefficientList[Series[Product[1/(1 + x^k)^4, {k, 1, nmax}], {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Aug 27 2015 *)

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

%Y Cf. A007259, A022597, A022569.

%Y Column k=4 of A286352.

%K sign

%O 0,2

%A _N. J. A. Sloane_