A001937 Expansion of (psi(x^2) / psi(-x))^3 in powers of x where psi() is a Ramanujan theta function.
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
Keywords
Examples
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 + ...
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
Programs
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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
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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 *) -
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 */
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
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
Extensions
Corrected and extended by Simon Plouffe
Checked and more terms from Olivier GERARD, May 06 2009
Comments