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A081360
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Expansion of q^(-1/24) (m (1-m) / 16)^(1/24) in powers of q, where m = k^2 is the parameter and q is the nome for Jacobian elliptic functions.
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17
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1, -1, 1, -2, 2, -3, 4, -5, 6, -8, 10, -12, 15, -18, 22, -27, 32, -38, 46, -54, 64, -76, 89, -104, 122, -142, 165, -192, 222, -256, 296, -340, 390, -448, 512, -585, 668, -760, 864, -982, 1113, -1260, 1426, -1610, 1816, -2048, 2304, -2590, 2910, -3264, 3658, -4097, 4582, -5120, 5718, -6378
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OFFSET
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0,4
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COMMENTS
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Number of partitions of n into distinct parts with an even number of odd parts minus partitions of n into distinct parts with an odd number of odd parts. G.f.: Product_{i=1..oo} (1+(-1)^i*x^i). - Jon Perry, Jun 04 2004
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LINKS
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FORMULA
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Expansion of 1 / chi(x) = chi(-x) / chi(-x^2) = f(x) / phi(x) = f(-x) / phi(-x^2) = psi(-x) / f(-x^2) = f(-x^2) / f(x) = f(-x^4) / psi(x) in powers of x where phi(), psi(), chi(), f() are Ramanujan theta functions.
Expansion of (lambda * (1 - lambda) / (16 * q))^(1/24) in powers of q where lambda is a modular elliptic function and q = exp(Pi i z) is the nome. - Michael Somos, Jul 19 2012
Expansion of q^(-1/24) * eta(q) * eta(q^4) / eta(q^2)^2 in powers of q.
Expansion of q^(-1/24) / f(t) in powers of q = exp(Pi i t) where f() is Weber's function.
Euler transform of period 4 sequence [-1, 1, -1, 0, ...].
Given g.f. A(x), B(x) = x * A(x^3)^8 satisfies 0 = f(B(x), B(x^2)) where f(u, v) = (u - v^2) * (v - u^2) - (4 * u * v * (1 - u*v))^2.
G.f. is a period 1 Fourier series which satisfies f(-1 / (2304 t)) = f(t) where q = exp(2 Pi i t). - Michael Somos, Jul 16 2007
G.f.: Product_{k>0} 1 / ( 1 + x^(2k - 1)) = Product_{k>0} (1 + (-x)^k).
a(n) ~ (-1)^n * exp(Pi*sqrt(n/3)) / (4*3^(1/4)*n^(3/4)). - Vaclav Kotesovec, Aug 30 2015
G.f.: (1/2)*(-1; -x)_inf, where (a; q)_inf is the q-Pochhammer symbol. - Vladimir Reshetnikov, Nov 21 2016
G.f.: exp(-Sum_{k>=1} x^k/(k*(1 - (-x)^k))). - Ilya Gutkovskiy, Jun 08 2018
Given g.f. A(x), B(x) = 2^(1/4) * x * A(x^24) satisfies 0 = f(B(x), B(x^5)) where f(u, v) = u^6 + v^6 + 2*u*v * ((u*v)^4 - 1). - Michael Somos, Mar 14 2019
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EXAMPLE
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G.f. = 1 - x + x^2 - 2*x^3 + 2*x^4 - 3*x^5 + 4*x^6 - 5*x^7 + 6*x^8 - 8*x^9 + ...
G.f. = q - q^25 + q^49 - 2*q^73 + 2*q^97 - 3*q^121 + 4*q^145 - 5*q^169 + ...
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MAPLE
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read theta; t1:=series(eta, q, 48); t2:= q^(-1/24)*t1*subs(q=q^4, t1)/subs(q=q^2, t1)^2; series(t2, q, 48); seriestolist(%); # N. J. A. Sloane, Aug 24 2007
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MATHEMATICA
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a[ n_] := SeriesCoefficient[ 1 / QPochhammer[ -x, x^2], {x, 0, n}]; (* Michael Somos, Jul 19 2012 *)
a[ n_] := SeriesCoefficient[ 1 / Product[ 1 + x^k, {k, 1, n, 2}], {x, 0, n}]; (* Michael Somos, Jul 19 2012 *)
a[ n_] := SeriesCoefficient[ With[ {m = ModularLambda[ Log[ q] / (Pi I)]}, ( m (1 - m) / (16 q))^(1/24)], {q, 0, n}]; (* Michael Somos, Jul 19 2012 *)
a[ n_] := SeriesCoefficient[ QPochhammer[ x, -x], {x, 0, n}]; (* Michael Somos, Nov 22 2016 *)
nmax = 100; CoefficientList[Series[Product[(1 + x^(2*k))/(1 + x^k), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Aug 30 2015 *)
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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 + A) * eta(x^4 + A) / eta(x^2 + A)^2, n))};
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CROSSREFS
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KEYWORD
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sign
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AUTHOR
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STATUS
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approved
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