A224884 Expansion of x / Series_Reversion(x*sqrt(1 + 4*x)).

1, 2, -6, 32, -210, 1536, -12012, 98304, -831402, 7208960, -63740820, 572522496, -5209363380, 47915728896, -444799488600, 4161823309824, -39209074920090, 371626340253696, -3541117629057540, 33902753847705600, -325969196485349340, 3146175557067079680, -30471769822097981160

Offset: 0

Author

Paul D. Hanna, Aug 21 2013

Keywords

• sign

Comments

Signed version of A206300. - Peter Bala, Mar 05 2020

Examples

			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 + ...
		

Links

• G. C. Greubel, Table of n, a(n) for n = 0..980

Crossrefs

Cf. A000108, A206300.

Cf. A000984, A048990, A078531, A182122, A245112, A245113.

Programs

Mathematica

CoefficientList[Series[x/InverseSeries[Series[x*Sqrt[1+4*x],{x,0,20}],x],{x,0,20}],x] (* Vaclav Kotesovec, Aug 22 2013 *)

PARI

{a(n)=polcoeff(x/serreverse(x*sqrt(1+4*x +x^2*O(x^n))),n)}
for(n=0,25,print1(a(n),", "))

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

G.f. A(x) satisfies A(x) = 1/A(-x/A(x)^4). - Seiichi Manyama, Jun 20 2025