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A100130
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Expansion of lambda * (1 - lambda) / 16 in powers of q.
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2
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1, -24, 300, -2624, 18126, -105504, 538296, -2471424, 10400997, -40674128, 149343012, -519045888, 1718732998, -5451292992, 16633756008, -49010118656, 139877936370, -387749049720, 1046413709980, -2754808758144, 7087483527072
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OFFSET
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1,2
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COMMENTS
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Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
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LINKS
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Table of n, a(n) for n=1..21.
M. Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
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FORMULA
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Expansion of q / chi(q)^24 in powers of q where chi() is a Ramanujan theta function.
Expansion of (eta(q) * eta(q^4) / eta(q^2)^2)^24 in powers of q.
Euler transform of period 4 sequence [ -24, 24, -24, 0, ...].
G.f. is a Fourier series which satisfies f(-1 / (4 t)) = f(t) where q = exp(2 pi i t).
G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = 4096 * (u*v)^4 + (u*v)^2 * (1791 + 2352 * (u + v) - 10496 * u*v) - u*v * (1 - 48 * (u + v) + 96 * (u^2 + v^2)) + u^3 + v^3.
G.f.: x * (Product_{k>0} (1 + (-x)^k))^24 = x / (Product_{k>0} (1 + x^(2*k - 1)))^24.
a(n) = -(-1)^n * A014103(n). Convolution inverse of A097340. Series inversion of A195130.
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EXAMPLE
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q - 24*q^2 + 300*q^3 - 2624*q^4 + 18126*q^5 - 105504*q^6 + 538296*q^7 - 2471424*q^8 + ...
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MATHEMATICA
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a[ n_] := With[ {m = InverseEllipticNomeQ @ q}, SeriesCoefficient[ (1 - m) m / 16, {q, 0, n}]]
a[ n_] := SeriesCoefficient[ x / Product[ 1 + x^k, {k, 1, n, 2}]^24, {x, 0, n}]
a[ n_] := SeriesCoefficient[ x / QPochhammer[ -x, x^2]^24, {x, 0, n}]
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PROG
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(PARI) {a(n) = polcoeff(x * prod(k=1, n, 1 + (-x)^k, 1 + x * O(x^n))^24, n)}
(PARI) {a(n) = local(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( (eta(x + A) * eta(x^4 + A) / eta(x^2 + A)^2)^24, n))}
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CROSSREFS
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Cf. A014103, A097340, A195130.
Sequence in context: A056285 A162686 A010976 * A014103 A206002 A000552
Adjacent sequences: A100127 A100128 A100129 * A100131 A100132 A100133
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KEYWORD
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sign
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AUTHOR
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Michael Somos, Nov 06 2004
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STATUS
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approved
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