A093160 Expansion of q^(-1/2) * (eta(q^4) / eta(q))^4 in powers of q.

1, 4, 14, 40, 101, 236, 518, 1080, 2162, 4180, 7840, 14328, 25591, 44776, 76918, 129952, 216240, 354864, 574958, 920600, 1457946, 2285452, 3548550, 5460592, 8332425, 12614088, 18953310, 28276968, 41904208, 61702876, 90304598

Offset: 0

Author

Michael Somos, Mar 26 2004, Apr 17 2007

Keywords

• nonn
• easy

Comments

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

Examples

			G.f. = 1 + 4*x + 14*x^2 + 40*x^3 + 101*x^4 + 236*x^5 + 518*x^6 + 1080*x^7 + ...
G.f. = q + 4*q^3 + 14*q^5 + 40*q^7 + 101*q^9 + 236*q^11 + 518*q^13 + ...
		

References

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

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..10000
• Michael Somos, Introduction to Ramanujan theta functions
• Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Crossrefs

Cf. A001938, A112143, A285927, A285928.

Programs

Mathematica

a[ n_] := SeriesCoefficient[ (Product[ 1 + x^k, {k, 2, n, 2}] / Product[ 1 - x^k, {k, 1, n, 2}])^4, {x, 0, n}];
a[ n_] := With[ {m = InverseEllipticNomeQ @ q}, SeriesCoefficient[ Sqrt[m] / (4 Sqrt[1 - m]), {q, 0, n + 1/2}]];
a[ n_] := With[ {m = InverseEllipticNomeQ @ q}, SeriesCoefficient[ m^(1/4) / (2 (1 - Sqrt @ m)), {q, 0, n/2 + 1/4 }]];
s = (QPochhammer[q^4]/QPochhammer[q])^4 + O[q]^30; CoefficientList[s, q] (* Jean-François Alcover, Nov 24 2015 *)

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))};

PARI

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

Formula

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

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

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.

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.

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

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.

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]

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

a(n) ~ exp(sqrt(2*n)*Pi) / (32 * 2^(1/4) * n^(3/4)). - Vaclav Kotesovec, Sep 07 2015

a(0) = 1, a(n) = (4/n)*Sum_{k=1..n} A046897(k)*a(n-k) for n > 0. - Seiichi Manyama, Apr 28 2017