|
| |
|
|
A014103
|
|
Expansion of (eta(q^2) / eta(q))^24 in powers of q.
|
|
8
|
|
|
|
1, 24, 300, 2624, 18126, 105504, 538296, 2471424, 10400997, 40674128, 149343012, 519045888, 1718732998, 5451292992, 16633756008, 49010118656, 139877936370, 387749049720, 1046413709980, 2754808758144, 7087483527072, 17848133716832
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
1,2
|
|
|
COMMENTS
|
Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
Given g.f. A(q), Greenhill (1895) denotes -64 * A(q^2) by tau_0 on page 409 equation (43). - Michael Somos, Jul 17 2013
|
|
|
REFERENCES
|
John H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 102.
|
|
|
LINKS
|
Seiichi Manyama, Table of n, a(n) for n = 1..10000
Kevin Acres, David Broadhurst, Eta quotients and Rademacher sums, arXiv:1810.07478 [math.NT], 2018. See Table 1 p. 10.
A. G. Greenhill, The Transformation and Division of Elliptic Functions, Proceedings of the London Mathematical Society (1895) 403-486.
R. S. Maier, On Rationally Parametrized Modular Equations, arXiv:math/0611041 [math.NT], 2006-2008, see page 4 equation (4)
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
Index entries for reversions of series
|
|
|
FORMULA
|
REVERT(A005149).
Euler transform of period 2 sequence [ 24, 0, 24, 0, ...]. - Michael Somos, Mar 19 2004
Expansion of (lambda(q) / 16)^2 / (1 - lambda(q)) in powers of q = exp(2 Pi i t) where lambda() is the elliptic modular function A115977. - Michael Somos, Nov 19 2005
Expansion of q / chi(-q)^24 in powers of q where chi() is a Ramanujan theta function.
Expansion of (theta_2(q) * theta_3(q) / (2 * theta_4(q)^2))^4 = (theta_2(q^(1/2))^2 / (4*theta_4(q^(1/2)) * theta_3(q^(1/2))))^4 in powers of q.
G.f.: x * Product_{k > 0} (1 + x^k)^24 = x / Product_{k > 0} (1 - x^(2*k - 1))^24.
G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = u^2 - v - 48*u*v - 4096*u*v^2. - Michael Somos, Mar 19 2004
G.f. is a period 1 Fourier series which satisfies f(-1 / (2 t)) = (1/4096) g(t) where q = exp(2 Pi i t) and g() is the g.f. of A007191. - Michael Somos, Aug 19 2007
j(q) = (f(q) + 16)^3 / f(q), j(q^2) = (f(q) + 256)^3 / f(q)^2 where j(q) is the g.f. for A000521 and f(q) is 4096 times the g.f. for a(n). - Michael Somos, Oct 01 2007
Convolution inverse of A007191. Series reversion of A005149.
Empirical : sum(exp(-2*Pi)^n*a(n), n = 1..infinity) = 1/512. - Simon Plouffe, Feb 20 2011
a(n) ~ exp(2 * Pi * sqrt(2*n)) / (4096 * 2^(3/4) * n^(3/4)). - Vaclav Kotesovec, Mar 05 2015
a(1) = 1, a(n) = (24/(n-1))*Sum_{k=1..n-1} A000593(k)*a(n-k) for n > 1. - Seiichi Manyama, Apr 01 2017
G.f.: x*exp(24*Sum_{k>=1} (-1)^(k+1)*x^k/(k*(1 - x^k))). - Ilya Gutkovskiy, Feb 06 2018
|
|
|
EXAMPLE
|
G.f. = q + 24*q^2 + 300*q^3 + 2624*q^4 + 18126*q^5 + 105504*q^6 + 538296*q^7 + ...
|
|
|
MAPLE
|
q*mul((1+q^m)^24, m=1..30);
|
|
|
MATHEMATICA
|
a[ n_] := SeriesCoefficient[ q QPochhammer[ q, q^2]^-24, {q, 0, n}]; (* Michael Somos, Jul 11 2011 *)
a[ n_] := SeriesCoefficient[ q / Product[ 1 - q^k, {k, 1, n + 1, 2}]^24, {q, 0, n}]; (* Michael Somos, Jul 11 2011 *)
a[ n_] := With[ {m = ModularLambda[ Log[q]/(Pi I)]}, SeriesCoefficient[ (m/16)^2 / (1 - m), {q, 0, 2 n}]]; (* Michael Somos, Jul 11 2011 *)
a[ n_] := With[ {m = InverseEllipticNomeQ @ q}, SeriesCoefficient[ (m/16)^2 /(1 - m), {q, 0, 2 n}]]; (* Michael Somos, Jul 11 2011 *)
eta[q_]:=q^(1/6) QPochhammer[q]; a[n_]:=SeriesCoefficient[(eta[q^2] / eta[q])^24, {q, 0, n}]; Table[a[n], {n, 4, 25}] (* Vincenzo Librandi, Oct 18 2018 *)
|
|
|
PROG
|
(PARI) {a(n) = polcoeff( x * prod( k=1, n, 1 + x^k, 1 + x * O(x^n))^24, n)};
(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( A + 16 * A^2, n))};
(PARI) {a(n) = my(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( (eta(x^2 + A) / eta(x + A))^24, n))};
|
|
|
CROSSREFS
|
Cf. A005149, A007191, A115977.
Sequence in context: A162686 A010976 A100130 * A321953 A206002 A000552
Adjacent sequences: A014100 A014101 A014102 * A014104 A014105 A014106
|
|
|
KEYWORD
|
nonn
|
|
|
AUTHOR
|
N. J. A. Sloane
|
|
|
EXTENSIONS
|
More terms from Michael Somos, Nov 24 2001
|
|
|
STATUS
|
approved
|
| |
|
|
