A093160 - OEIS

%I #31 Mar 12 2021 22:24:42

%S 1,4,14,40,101,236,518,1080,2162,4180,7840,14328,25591,44776,76918,

%T 129952,216240,354864,574958,920600,1457946,2285452,3548550,5460592,

%U 8332425,12614088,18953310,28276968,41904208,61702876,90304598

%N Expansion of q^(-1/2) * (eta(q^4) / eta(q))^4 in powers of q.

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

%D A. Cayley, An Elementary Treatise on Elliptic Functions, 2nd ed, 1895, p. 381, Section 488.

%H Seiichi Manyama, <a href="/A093160/b093160.txt">Table of n, a(n) for n = 0..10000</a>

%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>

%F G.f.: (Product_{k>0} (1 + x^(2*k)) / (1 - x^(2*k - 1)))^4.

%F Expansion of q^(-1/2) * k / (4 * k') in powers of q where q is Jacobi's nome and k is the elliptic modulus.

%F Expansion of q^(-1/4) * k^(1/2) / (2 * (1 - k)) in powers of q^(1/2) where q is Jacobi's nome and k is the elliptic modulus.

%F Expansion of (psi(x^2) / phi(-x))^2 = (psi(x) / phi(-x^2))^4 = (psi(-x) / phi(-x))^4 = (psi(x^2) / psi(-x))^4 = (chi(x) / chi(-x^2)^2)^4 = ( chi(x) * chi(-x)^2)^-4 = (chi(-x) * chi(-x^2))^-4 = (f(-x^4) / f(-x))^4 in powers of x where phi(), psi(), chi(), f() are Ramanujan theta functions.

%F Euler transform of period 4 sequence [ 4, 4, 4, 0, ...].

%F Given g.f. A(x), then B(x) = q * A(q^2) satisfies 0 = f(B(q), B(q^2)) where f(u, v) = u^2 - v - 16*u*v - 16*v^2 - 256*u*v^2.

%F G.f. A(q) satisfies A(q) = sqrt(A(-q^2)) / (1 - 4*q*A(-q^2)); together with limit_{n->infinity} A(x^n) = 1 this gives a fast algorithm to compute the series. [_Joerg Arndt_, Aug 06 2011]

%F A001938(n) = (-1)^n * a(n). Convolution inverse of A112143.

%F a(n) ~ exp(sqrt(2*n)*Pi) / (32 * 2^(1/4) * n^(3/4)). - _Vaclav Kotesovec_, Sep 07 2015

%F a(0) = 1, a(n) = (4/n)*Sum_{k=1..n} A046897(k)*a(n-k) for n > 0. - _Seiichi Manyama_, Apr 28 2017

%e G.f. = 1 + 4*x + 14*x^2 + 40*x^3 + 101*x^4 + 236*x^5 + 518*x^6 + 1080*x^7 + ...

%e G.f. = q + 4*q^3 + 14*q^5 + 40*q^7 + 101*q^9 + 236*q^11 + 518*q^13 + ...

%t a[ n_] := SeriesCoefficient[ (Product[ 1 + x^k, {k, 2, n, 2}] / Product[ 1 - x^k, {k, 1, n, 2}])^4, {x, 0, n}];

%t a[ n_] := With[ {m = InverseEllipticNomeQ @ q}, SeriesCoefficient[ Sqrt[m] / (4 Sqrt[1 - m]), {q, 0, n + 1/2}]];

%t a[ n_] := With[ {m = InverseEllipticNomeQ @ q}, SeriesCoefficient[ m^(1/4) / (2 (1 - Sqrt @ m)), {q, 0, n/2 + 1/4 }]];

%t s = (QPochhammer[q^4]/QPochhammer[q])^4 + O[q]^30; CoefficientList[s, q] (* _Jean-François Alcover_, Nov 24 2015 *)

%o (PARI) {a(n) = my(A, A2, m); if( n<0, 0, A = x + O(x^2); m=1; while( m<=n, m*=2; A = subst(A, x, x^2); A2 = A * (1 + 16*A); A = 8*A2 + (1 + 32*A) * sqrt(A2)); polcoeff( sqrt(A/x), n))};

%o (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x^4 + A) / eta(x + A))^4, n))};

%Y Cf. A001938, A112143, A285927, A285928.

%K nonn,easy

%O 0,2

%A _Michael Somos_, Mar 26 2004, Apr 17 2007