A022599 Expansion of Product_{m>=1} (1+q^m)^(-4).

1, -4, 6, -8, 17, -28, 38, -56, 84, -124, 172, -232, 325, -448, 594, -784, 1049, -1388, 1796, -2320, 3005, -3864, 4912, -6216, 7877, -9940, 12430, -15488, 19309, -23972, 29580, -36408, 44766, -54876, 66978, -81536, 99150, -120272, 145374, -175344, 211242

Offset: 0

Author

N. J. A. Sloane

Keywords

• sign

Comments

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

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

Examples

			G.f. = 1 - 4*x + 6*x^2 - 8*x^3 + 17*x^4 - 28*x^5 + 38*x^6 - 56*x^7 + ...
T12J = 1/q - 4*q^5 + 6*q^11 - 8*q^17 + 17*q^23 - 28*q^29 + 38*q^35 + ...
		

References

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

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..1000 from Alois P. Heinz)
• J. H. Conway and S. P. Norton, Monstrous Moonshine, Bull. Lond. Math. Soc. 11 (1979) 308-339.
• D. Ford, J. McKay and S. P. Norton, More on replicable functions, Commun. Algebra 22, No. 13, 5175-5193 (1994).
• Vaclav Kotesovec, A method of finding the asymptotics of q-series based on the convolution of generating functions, arXiv:1509.08708 [math.CO], Sep 30 2015, p. 13.
• J. McKay and H. Strauss, The q-series of monstrous moonshine and the decomposition of the head characters, Comm. Algebra 18 (1990), no. 1, 253-278.
• 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. A007259, A022597, A022569.

Column k=4 of A286352.

Programs

Maple

with(numtheory):
a:= proc(n) option remember; `if`(n=0, 1, add(add(
      -4*irem(d, 2)*d, d=divisors(j))*a(n-j), j=1..n)/n)
    end:
seq(a(n), n=0..50);  # Alois P. Heinz, May 02 2014

Mathematica

a[ n_] := SeriesCoefficient[ (QPochhammer[ x] / QPochhammer[x^2])^4, {x, 0, n}]; (* Michael Somos, Jul 05 2014 *)
nmax = 50; CoefficientList[Series[Product[1/(1 + x^k)^4, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Aug 27 2015 *)

PARI

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

Formula

Expansion of chi(-x)^4 in powers of x where chi() is a Ramanujan theta function.

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

Euler transform of period 2 sequence [ -4, 0, ...]. - Michael Somos, Apr 26 2008

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

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

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.

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

a(n) ~ (-1)^n * exp(Pi*sqrt(2*n/3)) / (2 * 6^(1/4) * n^(3/4)). - Vaclav Kotesovec, Aug 27 2015

a(0) = 1, a(n) = -(4/n)*Sum_{k=1..n} A000593(k)*a(n-k) for n > 0. - Seiichi Manyama, Apr 03 2017

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