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

%I A001937 M2785 N1120 #44 Feb 16 2025 08:32:24
%S A001937 1,3,9,22,48,99,194,363,657,1155,1977,3312,5443,8787,13968,21894,
%T A001937 33873,51795,78345,117312,174033,255945,373353,540486,776848,1109040,
%U A001937 1573209,2218198,3109713,4335840,6014123,8300811,11402928,15593702,21232521,28790667,38884082
%N A001937 Expansion of (psi(x^2) / psi(-x))^3 in powers of x where psi() is a Ramanujan theta function.
%C A001937 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%C A001937 The Cayley reference is actually to A187053. - _Michael Somos_, Jul 26 2012
%D A001937 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.
%D A001937 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D A001937 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A001937 Seiichi Manyama, <a href="/A001937/b001937.txt">Table of n, a(n) for n = 0..10000</a> (terms 0..1000 from T. D. Noe)
%H A001937 A. Cayley, <a href="/A001934/a001934.pdf">A memoir on the transformation of elliptic functions</a>, 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]
%H A001937 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A001937 Expansion of q^(-3/8) * (eta(q^4) / eta(q))^3 in powers of q. - _Michael Somos_, Jul 26 2012
%F A001937 Euler transform of period 4 sequence [ 3, 3, 3, 0, ...]. - _Michael Somos_, Mar 06 2011
%F A001937 Convolution cube of A001935. A187053(n) = (-1)^n * a(n). - _Michael Somos_, Mar 06 2011
%F A001937 G.f.: (Product_{k>0} (1 + x^(2*k)) / (1 - x^(2*k-1)))^3.
%F A001937 a(n) ~ 3^(1/4) * exp(sqrt(3*n/2)*Pi) / (16*2^(3/4)*n^(3/4)). - _Vaclav Kotesovec_, Nov 15 2017
%e A001937 1 + 3*x + 9*x^2 + 22*x^3 + 48*x^4 + 99*x^5 + 194*x^6 + 363*x^7 + 657*x^8 + ...
%e A001937 q^3 + 3*q^11 + 9*q^19 + 22*q^27 + 48*q^35 + 99*q^43 + 194*q^51 + 363*q^59 + ...
%p A001937 g100:= mul((1+x^(2*k))/(1-x^(2*k-1)),k=1..50)^3:
%p A001937 S:= series(g100,x,101):
%p A001937 seq(coeff(S,x,j),j=0..100); # _Robert Israel_, Nov 30 2015
%t A001937 CoefficientList[ Series[Product[(1 - x^k)^(-3*Boole[Mod[k, 4] != 0]), {k, 1, 101}], {x, 0, 100}], x] (* Olivier GERARD, May 06 2009 *)
%t A001937 QP = QPochhammer; s = (QP[q^4]/QP[q])^3 + O[q]^40; CoefficientList[s, q] (* _Jean-François Alcover_, Nov 30 2015, adapted from PARI *)
%o A001937 (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 */
%Y A001937 Cf. A001936, A079006, A001935, A083365, A001938, A093160, A001939, A001940, A001941.
%K A001937 nonn
%O A001937 0,2
%A A001937 _N. J. A. Sloane_, _Simon Plouffe_
%E A001937 Corrected and extended by _Simon Plouffe_
%E A001937 Checked and more terms from Olivier GERARD, May 06 2009