|
%I M4528 #90 Feb 23 2023 06:38:09
%S 0,1,-8,44,-192,718,-2400,7352,-20992,56549,-145008,356388,-844032,
%T 1934534,-4306368,9337704,-19771392,40965362,-83207976,165944732,
%U -325393024,628092832,-1194744096,2241688744,-4152367104,7599231223,-13749863984
%N Expansion of (theta_2(q)/theta_3(q))^4/16 in powers of q.
%C When multiplied by 16, this is the q-expansion of the automorphic function lambda (see A115977) [see Erdelyi].
%C Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%D M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math.Series 55, Tenth Printing, 1972, p. 591.
%D J. M. Borwein and P. B. Borwein, Pi and the AGM, Wiley, 1987, p. 121.
%D A. Erdelyi, Higher Transcendental Functions, McGraw-Hill, 1955, Vol. 3, p. 23, Eq. (37).
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H Seiichi Manyama, <a href="/A005798/b005798.txt">Table of n, a(n) for n = 0..10000</a> (terms 0..1000 from Alois P. Heinz)
%H M. Abramowitz and I. A. Stegun, eds., <a href="http://www.convertit.com/Go/ConvertIt/Reference/AMS55.ASP">Handbook of Mathematical Functions</a>, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
%H M. Abramowitz and I. A. Stegun, eds., <a href="http://www.convertit.com/Go/ConvertIt/Reference/AMS55.ASP">Handbook of Mathematical Functions</a>, National Bureau of Standards Applied Math.Series 55, Tenth Printing, 1972, p. 591.
%H A. Kneser, <a href="https://eudml.org/doc/149645">Neue Untersuchung einer Reihe aus der Theorie der elliptischen Funktionen</a>, J. reine u. angew. Math. 157, 1927, 209 - 218, p.213, second formula.
%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/EllipticLambdaFunction.html">Elliptic Lambda Function</a>.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>.
%F Expansion of elliptic lambda / 16 = m / 16 = (k / 4)^2 in powers of the nome q.
%F Expansion of q * (psi(q) / phi(q))^8 = q * (psi(q^2) / psi(q))^8 = q * (psi(-q) / phi(-q^2))^8 = q * (chi(-q) / chi(-q^2)^2)^8 = q / (chi(q) * chi(-q^2))^8 = q / (chi(-q) * chi(q)^2)^8 = q * (psi(q^2) / phi(q))^4 = q * (f(-q^4) / f(q))^8 in powers of q where phi(), psi(), chi(), f() are Ramanujan theta functions. - _Michael Somos_, Jun 13 2011
%F Expansion of eta(q)^8 * eta(q^4)^16 / eta(q^2)^24 in powers of q.
%F Euler transform of period 4 sequence [-8, 16, -8, 0, ...].
%F G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = u^2 - v + 16*u*v - 32*u^2*v + 256*(u*v)^2. - _Michael Somos_, Mar 19 2004
%F G.f. is a period 1 Fourier series which satisfies f(-1 / (4 t)) = (1 / 16) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A128692. - _Michael Somos_, May 10 2014
%F G.f.: q * Product( (1 + q^(2*n)) / (1 + q^(2*n - 1)), n=1..inf )^8.
%F a(n) ~ (-1)^(n+1) * exp(2*Pi*sqrt(n))/(512*n^(3/4)). - _Vaclav Kotesovec_, Jul 10 2016
%F Empirical: Sum_{n>=0} a(n)/exp(2*Pi*n) = 17/16 - 3*sqrt(2)/4, verified to 27000 digits (10000 terms). - _Simon Plouffe_, Mar 01 2021
%e G.f. = q - 8*q^2 + 44*q^3 - 192*q^4 + 718*q^5 - 2400*q^6 + 7352*q^7 - 20992*q^8 + ...
%p with (numtheory): etr:= proc(p) local b; b:=proc(n) option remember; local d,j; if n=0 then 1 else add (add (d*p(d), d=divisors(j)) *b(n-j), j=1..n)/n fi end end: aa:=etr (n-> [ -8,16,-8,0] [modp(n-1,4)+1]): a:= n->aa(n-1): seq (a(n), n=0..26); # _Alois P. Heinz_, Sep 08 2008
%t a[ n_] := SeriesCoefficient[ InverseEllipticNomeQ[ x] / 16, {x, 0, n}]; (* _Michael Somos_, Jun 13 2011 *)
%t a[ n_] := SeriesCoefficient[ (EllipticTheta[ 2, 0, q] / EllipticTheta[ 2, 0, q^(1/2)])^8, {q, 0, n}]; (* _Michael Somos_, May 10 2014 *)
%t a[ n_] := SeriesCoefficient[ q (QPochhammer[ q^4] / QPochhammer[ -q])^8, {q, 0, n}]; (* _Michael Somos_, May 10 2014 *)
%t a[ n_] := SeriesCoefficient[ q (Product[ 1 - q^k, {k, 4, n - 1, 4}] / Product[ 1 - (-q)^k, {k, n - 1}])^8, {q, 0, n}]; (* _Michael Somos_, May 10 2014 *)
%t etr[p_] := Module[{b}, b[n_] := b[n] = If[n == 0, 1, Sum[Sum[d*p[d], {d, Divisors[ j]}]*b[n-j], {j, 1, n}]/n]; b]; aa = etr[Function[{n}, {-8, 16, -8, 0}[[Mod[n-1, 4]+1]]]]; a[n_] := aa[n-1]; Table[a[n], {n, 0, 26}] (* _Jean-François Alcover_, Mar 05 2015, after _Alois P. Heinz_ *)
%o (PARI) {a(n) = my(A, m); if( n<1, 0, m=1; A = x + O(x^2); while( m<n, m*=2; A = sqrt( subst(A, x, x^2)); A /= (1 + 4*A)^2); polcoeff(A, n))};
%o (PARI) {a(n) = my(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( (eta(x + A) * eta(x^4 + A)^2 / eta(x^2 + A)^3)^8, n))}; /* _Michael Somos_, Jul 16 2005 */
%Y If initial 0 is omitted and sequence begins with a(0) = 1, then this is the convolution of A001938 with itself. G.f.s are related by a(x)=x*A001938(x)^2. Reversion of A005797.
%Y Cf. A007248, A029845, A115977, A128692.
%K sign,easy
%O 0,3
%A _N. J. A. Sloane_
%E Definition simplified by _N. J. A. Sloane_, Sep 25 2011
|
