A195861 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
Offset: 0
Keywords
Examples
G.f. = 1 - 5*x + 20*x^2 - 65*x^3 + 185*x^4 - 481*x^5 + 1165*x^6 - 2665*x^7 + ... G.f. = q^5 - 5*q^13 + 20*q^21 - 65*q^29 + 185*q^37 - 481*q^45 + 1165*q^53 - 2665*q^61 + ...
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.
Links
- Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..2500 from G. C. Greubel)
- 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
Crossrefs
Programs
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Mathematica
a[ n_] := With[ {m = InverseEllipticNomeQ @ q}, SeriesCoefficient[ (m / 16)^(5/8), {q, 0, n + 5/8}]]; a[ n_] := SeriesCoefficient[ Product[(1 + x^(k + 1)) / (1 + x^k), {k, 1, n, 2}]^5, {x, 0, n}]; a[ n_] := SeriesCoefficient[ (QPochhammer[ -x^2, x^2] / QPochhammer[ -x, x^2])^5, {x, 0, n}]; -
PARI
{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x + A) * eta(x^4 + A)^2 / eta(x^2 + A)^2)^5, n))};
Formula
Expansion of q^(-5/8) * (eta(q) * eta(q^4)^2 / eta(q^2)^3)^5 in powers of q.
Euler transform of period 4 sequence [-5, 10, -5, 0, ...].
G.f.: (Product_{k>0} (1 + x^(2*k)) / (1 + x^(2*k - 1)))^5.
a(n) ~ (-1)^n * 5^(1/4) * exp(sqrt(5*n/2)*Pi) / (64 * 2^(3/4) * n^(3/4)). - Vaclav Kotesovec, Nov 27 2015
Comments