A376229 G.f. A(x) satisfies A(x^2)^3/A(x^4)^3 = 1 + (A(x)/A(x^4) - 1)^2.

1, 3, 0, 3, 3, 0, 0, -3, 0, 21, 0, 0, 3, -39, 0, 0, 3, 45, 0, 141, 0, -210, 0, -450, 0, 1335, 0, -258, -3, -2385, 0, 1419, 0, -1161, 0, 15930, 21, -21702, 0, -50682, 0, 171396, 0, -137811, 0, -97470, 0, 339750, 3, -1146573, 0, 3472182, -39, -3829878, 0, -7427862, 0, 32038098, 0, -49562235, 0, 31304094, 0, 67155855, 3

Offset

0,2

Comments

Compare to the quadratic modular identity for the Jacobi theta_3 function,

(H(q)/H(q^4) - 1)^2 = H(q^2)^2/H(q^4)^2 - 1,

where H(q) = theta_3(q) = 1 + 2*Sum_{n>=1} x^(n^2) - see the Mathworld link.

Links

Paul D. Hanna, Table of n, a(n) for n = 0..5000

Weisstein, Eric W., Modular Equation. From MathWorld -- A Wolfram Web Resource.

Formula

G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies the following formulas.

(1) 1 = A(x^2)^3/A(x^4)^3 - (A(x)/A(x^4) - 1)^2.

(2) A(x^4) = (A(x^2)^3 - A(x^4)^3) / (A(x) - A(x^4))^2.

(3) A(x^4) = (A(x) + A(-x))/2.

(4) (A(x) - A(-x))/2 = sqrt( (A(x^2)^3 - A(x^4)^3)/A(x^4) ).

(5) A(x) = A(x^4) + sqrt( (A(x^2)^3 - A(x^4)^3)/A(x^4) ).

(6) A(-x) = A(x^4) - sqrt( (A(x^2)^3 - A(x^4)^3)/A(x^4) ).

(7) A(x)*A(-x) = A(x^4)^2 - (A(x) - A(x^4))^2.

Example

G.f.: A(x) = 1 + 3*x + 3*x^3 + 3*x^4 - 3*x^7 + 21*x^9 + 3*x^12 - 39*x^13 + 3*x^16 + 45*x^17 + 141*x^19 - 210*x^21 - 450*x^23 + 1335*x^25 + ...
RELATED SERIES.
A(x^2)^3/A(x^4)^3 = 1 + 9*x^2 + 18*x^4 - 45*x^6 - 126*x^8 + 297*x^10 + 720*x^12 - 1656*x^14 - 3654*x^16 + 8028*x^18 + ...
(A(x)/A(x^4) - 1)^2 = 9*x^2 + 18*x^4 - 45*x^6 - 126*x^8 + 297*x^10 + 720*x^12 - 1656*x^14 - 3654*x^16 + 8028*x^18 + ...
so that A(x^2)^3/A(x^4)^3 = 1 + (A(x)/A(x^4) - 1)^2.
A(x^2)/A(x^4) = 1 + 3*x^2 - 3*x^4 - 6*x^6 + 12*x^8 + 18*x^10 - 39*x^12 - 66*x^14 + 123*x^16 + 228*x^18 + ...
A(x)/A(x^4) = 1 + 3*x + 3*x^3 - 9*x^5 - 12*x^7 + 48*x^9 + 36*x^11 - 192*x^13 - 117*x^15 + 639*x^17 + ...
SPECIFIC VALUES.
A(z) = 0 at z = -0.31245635185437800233388341696108678693311070024380576235...
where A(z^4) = A(-z)/2 = 1.028596870899697277134090179343886902162374839735...
A(t) = -1 at t = -0.57039858230761517664247163660292675563165147675197101486...
where A(t^4) = (A(-t) - 1)/2 = 1.3215014702102533607352948276634473382263447772...
A(t) = 5 at t = 0.683825059088067195014401842872439819613739530174...
A(t) = 4 at t = 0.606471117891147006067772611853274691396307825562...
A(t) = 3 at t = 0.488969633907794978867963048254535917158223409614...
A(t) = 2 at t = 0.29877528015699666214801376809929520561876608507314234...
A(1/2) = 3.07658827064413627822065439440049423546493612856...
A(1/3) = 2.14782501592238791794931151802849624725193055717...
A(1/4) = 1.80849035462028760274165896397719191308936371488...
A(1/9) = 1.33790523401608922855357625459698254060987461490...
A(1/16) = 1.1882781873719088231283574549917342690921127491...
A(1/81) = 1.0370427517578948927021192402656783153702158307...
A(1/256) = 1.011718929512426215504567386284407285897386297...

Prog

(PARI) {a(n) = my(V=[1, 3], A); for(i=0, n, V = concat(V, 0); A = Ser(V);
V[#V] = polcoef( subst(A, x, x^2)^3/subst(A, x, x^4)^3 - 1 - (A/subst(A, x, x^4) - 1)^2, #V)/6 ); V[n+1]}
for(n=0, 60, print1(a(n), ", "))

Crossrefs

Sequence in context: A201582 A369881 A115379 * A127801 A096597 A097994

Adjacent sequences: A376226 A376227 A376228 * A376230 A376231 A376232

Keyword

sign

Author

Paul D. Hanna, Oct 15 2024

Status

approved