A001937 Expansion of (psi(x^2) / psi(-x))^3 in powers of x where psi() is a Ramanujan theta function. (Formerly M2785 N1120)

1, 3, 9, 22, 48, 99, 194, 363, 657, 1155, 1977, 3312, 5443, 8787, 13968, 21894, 33873, 51795, 78345, 117312, 174033, 255945, 373353, 540486, 776848, 1109040, 1573209, 2218198, 3109713, 4335840, 6014123, 8300811, 11402928, 15593702, 21232521, 28790667, 38884082

Offset

0,2

Comments

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

The Cayley reference is actually to A187053. - Michael Somos, Jul 26 2012

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]

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Formula

Expansion of q^(-3/8) * (eta(q^4) / eta(q))^3 in powers of q. - Michael Somos, Jul 26 2012

Euler transform of period 4 sequence [ 3, 3, 3, 0, ...]. - Michael Somos, Mar 06 2011

Convolution cube of A001935. A187053(n) = (-1)^n * a(n). - Michael Somos, Mar 06 2011

G.f.: (Product_{k>0} (1 + x^(2*k)) / (1 - x^(2*k-1)))^3.

a(n) ~ 3^(1/4) * exp(sqrt(3*n/2)*Pi) / (16*2^(3/4)*n^(3/4)). - Vaclav Kotesovec, Nov 15 2017

Example

1 + 3*x + 9*x^2 + 22*x^3 + 48*x^4 + 99*x^5 + 194*x^6 + 363*x^7 + 657*x^8 + ...
q^3 + 3*q^11 + 9*q^19 + 22*q^27 + 48*q^35 + 99*q^43 + 194*q^51 + 363*q^59 + ...

Maple

g100:= mul((1+x^(2*k))/(1-x^(2*k-1)), k=1..50)^3:
S:= series(g100, x, 101):
seq(coeff(S, x, j), j=0..100); # Robert Israel, Nov 30 2015

Mathematica

CoefficientList[ Series[Product[(1 - x^k)^(-3*Boole[Mod[k, 4] != 0]), {k, 1, 101}], {x, 0, 100}], x] (* Olivier GERARD, May 06 2009 *)
QP = QPochhammer; s = (QP[q^4]/QP[q])^3 + O[q]^40; CoefficientList[s, q] (* Jean-François Alcover, Nov 30 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))^3, n))} /* Michael Somos, Mar 06 2011 */

Crossrefs

Cf. A001936, A079006, A001935, A083365, A001938, A093160, A001939, A001940, A001941.

Sequence in context: A217878 A192389 A187053 * A086817 A247188 A000715

Adjacent sequences: A001934 A001935 A001936 * A001938 A001939 A001940

Keyword

nonn

Author

N. J. A. Sloane, Simon Plouffe

Extensions

Corrected and extended by Simon Plouffe

Checked and more terms from Olivier GERARD, May 06 2009

Status

approved