login
The OEIS is supported by the many generous donors to the OEIS Foundation.

 

Logo
Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A093160 Expansion of q^(-1/2) * (eta(q^4) / eta(q))^4 in powers of q. 19
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
COMMENTS
Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
REFERENCES
A. Cayley, An Elementary Treatise on Elliptic Functions, 2nd ed, 1895, p. 381, Section 488.
LINKS
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
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
EXAMPLE
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 + ...
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 *)
PROG
(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))};
CROSSREFS
Sequence in context: A331758 A291675 A066375 * A001938 A066368 A274327
KEYWORD
nonn,easy
AUTHOR
Michael Somos, Mar 26 2004, Apr 17 2007
STATUS
approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified March 29 02:23 EDT 2024. Contains 371264 sequences. (Running on oeis4.)