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A092877
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Expansion of (eta(q^4) / eta(q))^8 in powers of q.
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4
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1, 8, 44, 192, 718, 2400, 7352, 20992, 56549, 145008, 356388, 844032, 1934534, 4306368, 9337704, 19771392, 40965362, 83207976, 165944732, 325393024, 628092832, 1194744096, 2241688744, 4152367104, 7599231223, 13749863984
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,2
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COMMENTS
| Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
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LINKS
| Harry J. Smith, Table of n, a(n) for n=1,...,1000
M. Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
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FORMULA
| Expansion of q * (psi(-q) / phi(-q))^8 = q * (psi(q^2) / psi(-q))^8 = q * (psi(q) / phi(-q^2))^8 = q * (psi(q^2) / phi(-q))^4 = q * (chi(q) / chi(-q^2)^2)^8 = q / (chi(-q) * chi(-q^2))^8 = q / (chi(q) * chi(-q)^2)^8 = q * (f(-q^4) / f(-q))^8 in powers of q where phi(), psi(), chi(), f() are Ramanujan theta functions. - Michael Somos Jun 13 2011
Euler transform of period 4 sequence [ 8, 8, 8, 0, ...].
G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = u^2 - v - 16*u*v - 16*v^2 - 256*u*v^2.
G.f.: x * (Product_{k>0} (1 + x^(2*k)) / (1 - x^(2*k - 1)))^8.
G.f.: theta_2^4/(16*theta_4^4) = lambda / (16 * (1 - lambda)).
G.f.: exp( Integral theta_3(x)^4/x dx ). [From Paul D. Hanna (pauldhanan(AT)juno.com), May 03 2010]
A005798(n) = (-1)n * a(n).
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EXAMPLE
| q + 8*q^2 + 44*q^3 + 192*q^4 + 718*q^5 + 2400*q^6 + 7352*q^7 + 20992*q^8 + ...
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MATHEMATICA
| a[ n_] := If[ n < 0, 0, SeriesCoefficient[ -InverseEllipticNomeQ[ -x] / 16, {x, 0, n}]] (* Michael Somos Jun 13 2011 *)
a[ n_] := If[ n < 0, 0, SeriesCoefficient[ With[ {lambda = ModularLambda[ Log[x] / ( Pi I)]}, lambda / (16 * (1 - lambda))], {x, 0, n}]] (* Michael Somos Jun 13 2011 *)
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PROG
| (PARI) {a(n) = if( n<0, 0, polcoeff( x * prod(k=1, (n+1)\2, (1 + x^(2*k)) / (1 - x^(2*k-1)), 1 + x * O(x^n))^8, n))}
(PARI) {a(n) = local(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( (eta(x^4 + A) / eta(x + A))^8, n))}
(PARI) a(n)= { local(A); n--; A=x*O(x^n); polcoeff((eta(x^4 + A)/eta(x + A))^8, n); } { for(n=1, 1000, write("b092877.txt", n, " ", a(n)); ); } [From Harry J. Smith (hjsmithh(AT)sbcglobal.net), Jun 21 2009]
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CROSSREFS
| Cf. A005798.
Sequence in context: A059596 A181358 A005798 * A160521 A023007 A169795
Adjacent sequences: A092874 A092875 A092876 * A092878 A092879 A092880
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KEYWORD
| nonn
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AUTHOR
| Michael Somos, Mar 19 2004
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