A005798 Expansion of (theta_2(q)/theta_3(q))^4/16 in powers of q.

0, 1, -8, 44, -192, 718, -2400, 7352, -20992, 56549, -145008, 356388, -844032, 1934534, -4306368, 9337704, -19771392, 40965362, -83207976, 165944732, -325393024, 628092832, -1194744096, 2241688744, -4152367104, 7599231223, -13749863984

Offset: 0

Author

N. J. A. Sloane

Keywords

• sign
• easy

Comments

When multiplied by 16, this is the q-expansion of the automorphic function lambda (see A115977) [see Erdelyi].

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

Examples

			G.f. = q - 8*q^2 + 44*q^3 - 192*q^4 + 718*q^5 - 2400*q^6 + 7352*q^7 - 20992*q^8 + ...
		

References

• M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math.Series 55, Tenth Printing, 1972, p. 591.
• J. M. Borwein and P. B. Borwein, Pi and the AGM, Wiley, 1987, p. 121.
• A. Erdelyi, Higher Transcendental Functions, McGraw-Hill, 1955, Vol. 3, p. 23, Eq. (37).
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..1000 from Alois P. Heinz)
• M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
• M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math.Series 55, Tenth Printing, 1972, p. 591.
• A. Kneser, Neue Untersuchung einer Reihe aus der Theorie der elliptischen Funktionen, J. reine u. angew. Math. 157, 1927, 209 - 218, p.213, second formula.
• Michael Somos, Introduction to Ramanujan theta functions.
• Eric Weisstein's World of Mathematics, Elliptic Lambda Function.
• Eric Weisstein's World of Mathematics, Ramanujan Theta Functions.

Crossrefs

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.

Cf. A007248, A029845, A115977, A128692.

Programs

Maple

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

Mathematica

a[ n_] := SeriesCoefficient[ InverseEllipticNomeQ[ x] / 16, {x, 0, n}]; (* Michael Somos, Jun 13 2011 *)
a[ n_] := SeriesCoefficient[ (EllipticTheta[ 2, 0, q] / EllipticTheta[ 2, 0, q^(1/2)])^8, {q, 0, n}]; (* Michael Somos, May 10 2014 *)
a[ n_] := SeriesCoefficient[ q (QPochhammer[ q^4] / QPochhammer[ -q])^8, {q, 0, n}]; (* Michael Somos, May 10 2014 *)
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 *)
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 *)

PARI

{a(n) = my(A, m); if( n<1, 0, m=1; A = x + O(x^2); while( m
				

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 */

Formula

Expansion of elliptic lambda / 16 = m / 16 = (k / 4)^2 in powers of the nome q.

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

Expansion of eta(q)^8 * eta(q^4)^16 / eta(q^2)^24 in powers of q.

Euler transform of period 4 sequence [-8, 16, -8, 0, ...].

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

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

G.f.: q * Product( (1 + q^(2*n)) / (1 + q^(2*n - 1)), n=1..inf )^8.

a(n) ~ (-1)^(n+1) * exp(2*Pi*sqrt(n))/(512*n^(3/4)). - Vaclav Kotesovec, Jul 10 2016

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

Extensions

Definition simplified by N. J. A. Sloane, Sep 25 2011