A239112 Nonzero coefficients of g.f. A(x) = 1 + 4*x^2 + 36*x^4 + 396*x^6 + ... satisfying (A-1)*(1+3/A)^3 = 256*x^2.

1, 4, 36, 396, 4788, 61200, 811548, 11047176, 153362196, 2161860912, 30850416144, 444692531052, 6464058125868, 94633260282288, 1393927177591080, 20641691487933648, 307096967638064340, 4587717402640280304, 68788081908126980784, 1034809142155901808624

Offset: 0

Author

N. J. A. Sloane, Mar 29 2014

Keywords

• nonn

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..250
• A. Bostan, Computer Algebra for Lattice Path Combinatorics, Séminaire de Combinatoire Ph. Flajolet, March 28 2013. See Slide 13.
• M. Bousquet-Mélou, Plane lattice walks avoiding a quadrant, arXiv:1511.02111 [math.CO], 2015. See App. A.
• Mireille Bousquet-Mélou, Square lattice walks avoiding a quadrant, Journal of Combinatorial Theory, Series A, Elsevier, 2016, Special issue for the 50th anniversary of the journal, 144, pp. 37-79. Also . See App. A.
• V. Kotesovec, Asymptotic of implicit functions if Fww = 0

Programs

Mathematica

CoefficientList[Series[-2 + 64*x + Sqrt[1 - 256*x + 4096*x^2 + 12*(x - 16*x^2)^(1/3)] + (1/2)*Sqrt[-24 + 32*(1 - 32*x)^2 - 48*(x - 16*x^2)^(1/3) + (8*(1 + 480*x - 24576*x^2 + 262144*x^3)) / Sqrt[1 - 256*x + 4096*x^2 + 12*(x - 16*x^2)^(1/3)]], {x, 0, 20}], x] (* Vaclav Kotesovec, Mar 30 2014 *)
Flatten[{1,RecurrenceTable[{256 (-3+n) (-5+3 n) (-1+3 n) a[-2+n]-8 (-15+82 n-108 n^2+36 n^3) a[-1+n]+n (-2+3 n) (-1+3 n) a[n]==0, a[1]==4, a[2]==36}, a, {n,20}]}] (* Vaclav Kotesovec, Mar 30 2014 *)
terms = 20; B[] = 1; Do[B[x] = 1+256*x/(1+3/B[x])^3 + O[x]^terms, terms]; CoefficientList[B[x], x] (* Jean-François Alcover, Jan 15 2018, after Vaclav Kotesovec *)

PARI

{a(n)=local(B=1+x*O(x^n)); for(i=0, n, B=1+256*x/(1+3/B)^3); polcoeff(B, n)}
for(n=0, 20, print1(a(n), ", ")) \\ Vaclav Kotesovec, Mar 30 2014

Formula

From Vaclav Kotesovec, Mar 30 2014: (Start)

G.f. B(x) satisfies: B(x) = 1+256*x/(1+3/B(x))^3.

G.f.: -2 + 64*x + sqrt(1 - 256*x + 4096*x^2 + 12*(x - 16*x^2)^(1/3)) + sqrt(-24 + 32*(1 - 32*x)^2 - 48*(x - 16*x^2)^(1/3) + (8*(1 + 480*x - 24576*x^2 + 262144*x^3))/sqrt(1 - 256*x + 4096*x^2 + 12*(x - 16*x^2)^(1/3)))/2.

D-finite with recurrence n*(3*n-2)*(3*n-1)*a(n) = 8*(36*n^3 - 108*n^2 + 82*n - 15)*a(n-1) - 256*(n-3)*(3*n-5)*(3*n-1)*a(n-2).

a(n) ~ sqrt(3) * GAMMA(1/3) * 2^(4*n-1/3) / (Pi * n^(4/3)) * (1 - 3*2^(1/3) * sqrt(Pi) / (n^(1/3) * GAMMA(1/6))).

(End)

Extensions

a(7)-a(20) from Vaclav Kotesovec, Mar 30 2014