A001939 Expansion of (psi(-x) / phi(-x))^5 in powers of x where phi(), psi() are Ramanujan theta functions.
1, 5, 20, 65, 185, 481, 1165, 2665, 5820, 12220, 24802, 48880, 93865, 176125, 323685, 583798, 1035060, 1806600, 3108085, 5276305, 8846884, 14663645, 24044285, 39029560, 62755345, 100004806, 158022900, 247710570, 385366265, 595212280, 913040649, 1391449780
Offset: 0
Examples
1 + 5*x + 20*x^2 + 65*x^3 + 185*x^4 + 481*x^5 + 1165*x^6 + 2665*x^7 + ... q^5 + 5*q^13 + 20*q^21 + 65*q^29 + 185*q^37 + 481*q^45 + 1165*q^53 + ...
References
- 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.
- N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
- N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
Links
- Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..1000 from T. D. Noe)
- A. Cayley, A memoir on the transformation of elliptic functions, 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]
- Michael Somos, Introduction to Ramanujan theta functions
- Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
Programs
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Mathematica
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 *) 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 *) 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 *) QP = QPochhammer; s = (QP[q^4]/QP[q])^5 + O[q]^40; CoefficientList[s, q] (* Jean-François Alcover, Nov 27 2015, adapted from PARI *) -
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 */
Formula
Expansion of q^(-5/8) * (eta(q^4) / eta(q))^5 in powers of q. - Michael Somos, Sep 24 2011
Euler transform of period 4 sequence [ 5, 5, 5, 0, ...]. - Michael Somos, Sep 24 2011
G.f.: (Product_{k>0} (1 - x^(4*k)) / (1 - x^k))^5. - Michael Somos, Sep 24 2011
a(n) = (-1)^n * A195861(n). - Michael Somos, Sep 24 2011
a(n) ~ 5^(1/4) * exp(sqrt(5*n/2)*Pi) / (64 * 2^(3/4) * n^(3/4)). - Vaclav Kotesovec, Nov 27 2015
Comments