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A029839
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McKay-Thompson series of class 16B for the Monster group.
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21
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1, 2, -1, -2, 3, 2, -4, -4, 5, 8, -8, -10, 11, 12, -15, -18, 22, 26, -29, -34, 38, 42, -51, -56, 66, 78, -85, -98, 109, 120, -139, -156, 176, 202, -222, -250, 279, 306, -346, -384, 429, 482, -530, -590, 650, 714, -797, -876, 972, 1080, -1180, -1304, 1431, 1562, -1728, -1892, 2078, 2290, -2496
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OFFSET
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0,2
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COMMENTS
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In [Klein and Fricke 1890], the g.f. A(q)/2 is denoted by mu. On page 613 special values given are mu(i infinity) = infinity, mu(0) = 1, mu(2) = -1 and on page 615 properties given are mu(omega+1) = -i mu(omega), mu(-1/omega) = (mu(omega)+1)/(mu(omega)-1). - Michael Somos, Nov 09 2014
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LINKS
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FORMULA
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Expansion of q times normalized Hauptmodul for Gamma(4) in powers of q^4.
Expansion of q^(1/4) * eta(q^2)^6 / (eta(q)^2 * eta(q^4)^4) in powers of q.
Euler transform of period 4 sequence [2, -4, 2, 0, ...].
G.f. A(x) satisfies: A(x)^2 = A(x^2) + 4*x / A(x^2). - Michael Somos, Mar 08 2004
G.f.: Product_{k>0} ((1 + x^(2*k-1)) / (1 + x^(2*k)))^2.
Given g.f. A(x), then B(q) = A(q^4) / q satisfies 0 = f(B(q), B(q^2)) where f(u, v) = 4 + v^2 - u^2*v. - Michael Somos, May 14 2004
Given g.f. A(x), then B(q) = A(q^4) / (2*q) satisfies 0 = f(B(q), B(q^3)) where f(u, v) = (1 - u^4) * (1 - v^4) - (1 - u*v)^4. - Michael Somos, Oct 04 2006
Given g.f. A(x), then B(q) = A(q^4) / q satisfies 0 = f(B(q), B(q^2), B(q^3), B(q^6)) where f(u1, u2, u3, u6) = (u6 + u2)^2 - u1*u2*u3*u6. - Michael Somos, Oct 04 2006
Expansion of q^(1/4) * 2 / k(q)^(1/2) in powers of Jacobi nome q where k() is the elliptic modulus.
Expansion of q^(1/2) * 2 * (1 + k'(q)) / k(q) in powers of q^2. - Michael Somos, Nov 09 2014
Expansion of phi(x) / psi(x^2) = phi(x)^2 / psi(x)^2 = psi(x)^2 / psi(x^2)^2 = phi(-x^2)^2 / psi(-x)^2 = chi(-x^2)^4 / chi(-x)^2 = chi(x)^2 * chi(-x^2)^2 = chi(x)^4 * chi(-x)^2 = f(x)^2 / f(-x^4)^2 in powers of x where phi(), psi(), chi(), f() are Ramanujan theta functions.
Expansion of continued fraction 1 - x^2 + (x^1 + x^3)^2 / (1 - x^6 + (x^2 + x^6)^2 / (1 - x^10 + (x^3 + x^9)^2 / ...)) in powers of x^4. - Michael Somos, Apr 27 2008
G.f. is a period 1 Fourier series which satisfies f(-1 / (16 t)) = 2 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A007096.
A(q) = Sum_{n = -oo..oo} q^n/(1 - q^(4*n+1)) / Sum_{n = -oo..oo} q^(2*n)/(1 - q^(4*n+1)).
A(q) = ( 1 + q/(1 + (q + q^2)/(1 + q^3/(1 + (q^2 + q^4)/(1 + q^5/(1 + ... ))))) )^2. See Agarwal, p. 285.
A(q) = B(q)^2, where B(q) is the g.f. of A029838. (End)
abs(a(n)) ~ exp(Pi*sqrt(n)/2) / (2^(3/2) * n^(3/4)). - Vaclav Kotesovec, Feb 07 2023
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EXAMPLE
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G.f. = 1 + 2*x - x^2 - 2*x^3 + 3*x^4 + 2*x^5 - 4*x^6 - 4*x^7 + 5*x^8 + 8*x^9 + ...
T16B = 1/q + 2*q^3 - q^7 - 2*q^11 + 3*q^15 + 2*q^19 - 4*q^23 - 4*q^27 + ...
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MATHEMATICA
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a[0] = 1; a[n_] := Module[{A, m}, If[n < 0, 0, A = 1; m = 1; While[m <= n, m *= 2; A = A /. x -> x^2; A = Sqrt[A + 4*x/A]]; SeriesCoefficient[A, {x, 0, n}]]]; Table[a[n], {n, 0, 58}] (* Jean-François Alcover, Mar 12 2014, after PARI *)
a[ n_] := SeriesCoefficient[ 2 q^(1/4) EllipticTheta[ 3, 0, q] / EllipticTheta[ 2, 0, q], {q, 0, n}]; (* Michael Somos, Jul 05 2014 *)
QP = QPochhammer; s = QP[q^2]^6/(QP[q]^2*QP[q^4]^4) + O[q]^60; CoefficientList[s, q] (* Jean-François Alcover, Nov 16 2015, adapted from PARI *)
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PROG
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(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x^2 + A)^3 / (eta(x + A) * eta(x^4 + A)^2))^2, n))};
(PARI) {a(n) = my(A, m); if( n<0, 0, A = 1 + O(x); m=1; while( m<=n, m*=2; A = subst(A, x, x^2); A = sqrt(A + 4*x/A)); polcoeff(A, n))};
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CROSSREFS
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KEYWORD
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sign,easy
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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