OFFSET
0,2
COMMENTS
Signed version of A206300. - Peter Bala, Mar 05 2020
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..980
FORMULA
G.f. A(x) satisfies:
(1) A(x) = A(x)^3 - 4*x.
(2) A(x) = sqrt(1 + 4*x/A(x)).
(3) A(x*sqrt(1+4*x)) = sqrt(1+4*x).
(4) [x^n] A(x)^(n+2*k) = 0 for k=1..n-1, for n >= 2.
From Vaclav Kotesovec, Aug 22 2013: (Start)
a(n) = (-1)^(n+1) * 3^(3*n/2-1) * 4^(n-1) * GAMMA(n/2 - 1/6) * GAMMA(n/2 + 1/6)/(Pi*n!).
|a(n)| ~ 6^(n-1)*3^(n/2)/(sqrt(Pi/2)*n^(3/2)).
D-finite with recurrence: (n-1)*n*a(n) = 12*(3*n-7)*(3*n-5)*a(n-2). (End)
G.f.: (2/sqrt(3))*cosh(1/3*arccosh(sqrt(108)*x)). - Vladimir Kruchinin, Oct 11 2022
EXAMPLE
G.f.: A(x) = 1 + 2*x - 6*x^2 + 32*x^3 - 210*x^4 + 1536*x^5 - 12012*x^6 + ..
The coefficients in the powers A(x)^n of the g.f. begin:
n= 1: [1, 2, -6, 32, -210, 1536,-12012, 98304, -831402, ...];
n= 2: [1, 4, -8, 40, -256, 1848,-14336, 116688, -983040, ...];
n= 3: [1, 6, -6, 32, -210, 1536,-12012, 98304, -831402, ...];
n= 4: [1, 8, 0, 16, -128, 1008, -8192, 68640, -589824, ...];
n= 5: [1, 10, 10, 0, -50, 512, -4620, 40960, -364650, ...];
n= 6: [1, 12, 24, -8, 0, 168, -2048, 20592, -196608, ...];
n= 7: [1, 14, 42, 0, 14, 0, -588, 8192, -90090, ...];
n= 8: [1, 16, 64, 32, 0, -32, 0, 2112, -32768, ...];
n= 9: [1, 18, 90, 96, -18, 0, 84, 0, -7722, ...];
n=10: [1, 20, 120, 200, 0, 24, 0, -240, 0, ...];
n=11: [1, 22, 154, 352, 110, 0, -44, 0, 726, ...];
n=12: [1, 24, 192, 560, 384, -48, 0, 96, 0, ...];
n=13: [1, 26, 234, 832, 910, 0, 52, 0, -234, ...];
n=14: [1, 28, 280, 1176, 1792, 392, 0, -80, 0, ...];
n=15: [1, 30, 330, 1600, 3150, 1536, -140, 0, 150, ...];
n=16: [1, 32, 384, 2112, 5120, 4032, 0, 128, 0, ...];
n=17: [1, 34, 442, 2720, 7854, 8704, 1428, 0, -170, ...];
n=18: [1, 36, 504, 3432, 11520, 16632, 6144, -432, 0, ...];
n=19: [1, 38, 570, 4256, 16302, 29184, 17556, 0, 342, ...];
n=20: [1, 40, 640, 5200, 22400, 48048, 40960, 5280, 0, ...]; ...
which illustrates the property [x^n] A(x)^(n+2*k) = 0 for k=1..n-1:
[x^2] A(x)^4 = 0;
[x^3] A(x)^5 = 0, [x^3] A(x)^7 = 0;
[x^4] A(x)^6 = 0, [x^4] A(x)^8 = 0, [x^4] A(x)^10 = 0; ...
[x^5] A(x)^7 = 0, [x^5] A(x)^9 = 0, [x^5] A(x)^11 = 0, [x^5] A(x)^13 = 0; ...
Related series:
sqrt(1+4*x) = 1 + 2*x - 2*x^2 + 4*x^3 - 10*x^4 + 28*x^5 - 84*x^6 + 264*x^7 - 858*x^8 + ... + (-1)^(n-1)*2*A000108(n-1)*x^n + ...
MATHEMATICA
CoefficientList[Series[x/InverseSeries[Series[x*Sqrt[1+4*x], {x, 0, 20}], x], {x, 0, 20}], x] (* Vaclav Kotesovec, Aug 22 2013 *)
PROG
(PARI) {a(n)=polcoeff(x/serreverse(x*sqrt(1+4*x +x^2*O(x^n))), n)}
for(n=0, 25, print1(a(n), ", "))
CROSSREFS
KEYWORD
sign
AUTHOR
Paul D. Hanna, Aug 21 2013
STATUS
approved