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%I M3898 N1599 #45 Mar 12 2021 22:24:41
%S 1,5,20,65,185,481,1165,2665,5820,12220,24802,48880,93865,176125,
%T 323685,583798,1035060,1806600,3108085,5276305,8846884,14663645,
%U 24044285,39029560,62755345,100004806,158022900,247710570,385366265,595212280,913040649,1391449780
%N Expansion of (psi(-x) / phi(-x))^5 in powers of x where phi(), psi() are Ramanujan theta functions.
%C Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%D A. Cayley, A memoir on the transformation of elliptic functions, Collected Mathematical Papers. Vols. 1-13, Cambridge Univ. Press, London, 1889-1897, Vol. 9, p. 128.
%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H Seiichi Manyama, <a href="/A001939/b001939.txt">Table of n, a(n) for n = 0..10000</a> (terms 0..1000 from T. D. Noe)
%H A. Cayley, <a href="/A001934/a001934.pdf">A memoir on the transformation of elliptic functions</a>, Philosophical Transactions of the Royal Society of London (1874): 397-456; Collected Mathematical Papers. Vols. 1-13, Cambridge Univ. Press, London, 1889-1897, included in Vol. 9. [Annotated scan of pages 126-129]
%H Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F Expansion of q^(-5/8) * (eta(q^4) / eta(q))^5 in powers of q. - _Michael Somos_, Sep 24 2011
%F Euler transform of period 4 sequence [ 5, 5, 5, 0, ...]. - _Michael Somos_, Sep 24 2011
%F G.f.: (Product_{k>0} (1 - x^(4*k)) / (1 - x^k))^5. - _Michael Somos_, Sep 24 2011
%F a(n) = (-1)^n * A195861(n). - _Michael Somos_, Sep 24 2011
%F a(n) ~ 5^(1/4) * exp(sqrt(5*n/2)*Pi) / (64 * 2^(3/4) * n^(3/4)). - _Vaclav Kotesovec_, Nov 27 2015
%e 1 + 5*x + 20*x^2 + 65*x^3 + 185*x^4 + 481*x^5 + 1165*x^6 + 2665*x^7 + ...
%e q^5 + 5*q^13 + 20*q^21 + 65*q^29 + 185*q^37 + 481*q^45 + 1165*q^53 + ...
%t a[ n_] := SeriesCoefficient[ (EllipticTheta[ 2, 0, q] / EllipticTheta[ 2, Pi/4, q^(1/2)] / (16 q)^(1/8))^5, {q, 0, n}] (* _Michael Somos_, Sep 24 2011 *)
%t a[ n_] := SeriesCoefficient[ (Product[1 - x^k, {k, 4, n, 4}] / Product[1 - x^k, {k, n}])^5, {x, 0, n}] (* _Michael Somos_, Sep 24 2011 *)
%t nn = 4*20; b = Flatten[Table[{5, 5, 5, 0}, {nn/4}]]; CoefficientList[x*Series[Product[1/(1 - x^m)^b[[m]], {m, nn}], {x, 0, nn}], x] (* _T. D. Noe_, Aug 17 2012 *)
%t QP = QPochhammer; s = (QP[q^4]/QP[q])^5 + O[q]^40; CoefficientList[s, q] (* _Jean-François Alcover_, Nov 27 2015, adapted from PARI *)
%o (PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x^4 + A) / eta(x + A))^5, n))} /* _Michael Somos_, Sep 24 2011 */
%Y Cf. A000122, A000700, A010054, A121373, A195861.
%K nonn,easy
%O 0,2
%A _N. J. A. Sloane_, _Simon Plouffe_
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