|
%I #45 Feb 07 2023 12:25:35
%S 1,1,-1,0,1,0,-1,-1,2,1,-2,-1,2,1,-3,-1,4,2,-5,-2,5,2,-6,-3,8,4,-9,-4,
%T 10,4,-12,-6,15,7,-17,-7,19,8,-22,-10,26,12,-30,-13,33,14,-38,-17,45,
%U 21,-51,-22,56,24,-64,-29,74,33,-83,-36,92,40,-104,-46,119,53,-133,-58,147,63,-165,-73,187,83,-208,-90,229,99,-256
%N Expansion of square root of q times normalized Hauptmodul for Gamma(4) in powers of q^8.
%C Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%D B. C. Berndt, Ramanujan's Notebooks Part III, Springer-Verlag, see p. 221 Entry 1(i).
%H Seiichi Manyama, <a href="/A029838/b029838.txt">Table of n, a(n) for n = 0..10000</a>
%H W. Duke, <a href="http://dx.doi.org/10.1090/S0273-0979-05-01047-5">Continued fractions and modular functions</a>, Bull. Amer. Math. Soc. 42 (2005), 137-162; see Eqs. (9.1),(9.3).
%H J. McKay and A. Sebbar, <a href="http://dx.doi.org/10.1007/s002080000116">Fuchsian groups, automorphic functions and Schwarzians</a>, Math. Ann. 318 (2000), no. 2, 255-275.
%H Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>, <a href="http://mathworld.wolfram.com/q-PochhammerSymbol.html">q-Pochhammer Symbol</a>
%F Expansion of f(x) / f(-x^4) = phi(x) / psi(x) = psi(x) / psi(x^2) = phi(-x^2) / psi(-x) = chi(x) * chi(-x^2) = chi^2(x) * chi(-x) = chi^2(-x^2) / chi(-x) = (phi(x) / psi(x^2))^(1/2) in powers of x where phi(), psi(), chi(), f() are Ramanujan theta functions.
%F Expansion of q^(1/8) * eta(q^2)^3 / (eta(q) * eta(q^4)^2) in powers of q.
%F Euler transform of period 4 sequence [ 1, -2, 1, 0, ...].
%F Given g.f. A(x), then B(q) = A(q^8) / q satisfies 0 = f(B(q), B(q^2)) where f(u, v) = 4 + v^4 - u^4*v^2. - _Michael Somos_, Mar 02 2006
%F Given g.f. A(x), then B(q) = A(q^8) / q satisfies 0 = f(B(q), B(q^3)) where f(u, v) = u^4 - v^4 - 4*u*v + u^3*v^3. - _Michael Somos_, Mar 02 2006
%F Given g.f. A(x), then B(q) = A(q^8) / q satisfies 0 = f(B(q), B(q^2), B(q^4)) where f(u, v, w) = 2 + w^2 - u^2*v*w. - _Michael Somos_, Mar 02 2006
%F Given g.f. A(x), then B(q) = A(q^8) / q satisfies 0 = f(B(q), B(q^2), B(q^3), B(q^6)) where f(u1, u2, u3, u6) = u2^2 + u6^2 - u1*u2*u3*u6. - _Michael Somos_, Mar 02 2006
%F G.f. A(x) satisfies A(x)^2 = (A(x^4) + 2*x / A(x^4)) / A(x^2). - _Michael Somos_, Mar 08 2004
%F G.f. A(x) satisfies A(x) = (A(x^2)^2+4*x/A(x^2)^2)^(1/4). - _Joerg Arndt_, Aug 06 2011
%F G.f.: Product_{k>0} (1 + x^(2*k - 1)) / (1 + x^(2*k)) = (Sum_{k>0} x^((k^2 - k)/2)) / (Sum_{k>0} x^(k^2 - k)).
%F G.f.: 1 + x / (1 + x + x^2 / (1 + x^2 + x^3 / (1 + x^3 + ...))).
%F A082303(n) = (-1)^n a(n). Convolution square is A029839. Convolution inverse is A083365.
%F G.f.: 2 - 2/(1+Q(0)), where Q(k)= 1 + x^(k+1) + x^(k+1)/Q(k+1); (continued fraction). - _Sergei N. Gladkovskii_, May 02 2013
%F G.f.: (-x; x^2)_{1/2}, where (a; q)_n is the q-Pochhammer symbol. - _Vladimir Reshetnikov_, Nov 20 2016
%F a(0) = 1, a(n) = (1/n)*Sum_{k=1..n} A109506(k)*a(n-k) for n > 0. - _Seiichi Manyama_, Apr 14 2017
%F abs(a(n)) ~ sqrt(sqrt(2) + (-1)^n) * exp(Pi*sqrt(n)/2^(3/2)) / (4*n^(3/4)). - _Vaclav Kotesovec_, Feb 07 2023
%e G.f. = 1 + x - x^2 + x^4 - x^6 - x^7 + 2*x^8 + x^9 - 2*x^10 - x^11 + 2*x^12 + ...
%e G.f. = 1/q + q^7 - q^15 + q^31 - q^47 - q^55 + 2*q^63 + q^71 - 2*q^79 - q^87 + ...
%t a[ n_] := SeriesCoefficient[ QPochhammer[ -q, q^2] QPochhammer[ q^2, q^4], {q, 0, n}]; (* _Michael Somos_, Aug 20 2014 *)
%t a[ n_] := SeriesCoefficient[ QPochhammer[ -q, -q] / QPochhammer[ q^4, q^4], {q, 0, n}]; (* _Michael Somos_, Aug 20 2014 *)
%t a[ n_] := SeriesCoefficient[ q^(1/8) EllipticTheta[ 2, 0, q^(1/2)] / EllipticTheta[ 2, 0, q], {q, 0, n}]; (* _Michael Somos_, Aug 20 2014 *)
%t (QPochhammer[-x, x^2, 1/2] + O[x]^100)[[3]] (* _Vladimir Reshetnikov_, Nov 20 2016 *)
%o (PARI) {a(n) = if( n<0, 0, polcoeff( prod( k=1, n, (1 + x^k)^(-(-1)^k), 1 + x * O(x^n)), n))};
%o (PARI) {a(n) = my(A); if( n<0, 0, A = contfracpnqn( matrix(2, (sqrtint(8*n + 1) + 1)\2, i, j, if( i==1, x^(j-1), 1 + if( j>1, x^(j-1))))); polcoeff(A[1,1] / A[2,1] + x * O(x^n), n))}; /* _Michael Somos_, Mar 02 2006 */
%o (PARI) {a(n) = my(A, m); if( n<0, 0, A = 1 + O(x); m=1; while( m<=n, m*=2; A = subst(A, x, x^2); A2 = subst(A, x, x^2); A = sqrt((A2 + 2 * x / A2) / A)); polcoeff(A, n))};
%o (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^3 / eta(x + A) / eta(x^4 + A)^2, n))};
%Y Cf. A029839, A082303, A083365.
%K sign
%O 0,9
%A _N. J. A. Sloane_
|
