A001939 Expansion of (psi(-x) / phi(-x))^5 in powers of x where phi(), psi() are Ramanujan theta functions. (Formerly M3898 N1599)

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

0,2

Comments

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

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)

Arthur 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]

Simon Plouffe, Numbers in the base e^Pi, arXiv:2509.15609 [math.NT], 2025. See p. 15/24, marked 12.

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

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

Empirical: Sum_{n>=0} a(n) / exp(n*Pi) = (1/8) * exp(5*Pi/8) * sqrt(2) = A388098. - Simon Plouffe, Sep 14 2025

Example

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 + ...

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 *)

Prog

(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 */

Crossrefs

Cf. A000122, A000700, A010054, A121373, A195861.

Sequence in context: A271411 A309919 A195861 * A100534 A285928 A160506

Adjacent sequences: A001936 A001937 A001938 * A001940 A001941 A001942

Keyword

nonn,easy

Author

N. J. A. Sloane, Simon Plouffe

Status

approved