A120606 - OEIS

A120606

G.f. satisfies: 36*A(x) = 35 + 81*x + A(x)^9, starting with [1,3,12].

1, 3, 12, 180, 3018, 56238, 1121484, 23406804, 504914175, 11167352013, 251879507880, 5771456609880, 133970974830420, 3143760834627420, 74454455230816008, 1777349666975945784, 42721359085344132657, 1033093137613339252467, 25116105553098288701700

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OFFSET

0,2

COMMENTS

See comments in A120588 for conditions needed for an integer sequence to satisfy a functional equation of the form: r*A(x) = c + b*x + A(x)^n.

LINKS

Robert Israel, Table of n, a(n) for n = 0..703

FORMULA

G.f.: A(x) = 1 + Series_Reversion((1+36*x - (1+x)^9)/81). Lagrange Inversion yields: G.f.: A(x) = Sum_{n>=0} C(9*n,n)/(8*n+1) * (35+81*x)^(8*n+1)/36^(9*n+1). - Paul D. Hanna, Jan 24 2008

a(n) ~ 3^(-1 + 4*n) * (-35 + 2^(21/4))^(1/2 - n) / (2^(23/8) * n^(3/2) * sqrt(Pi)). - Vaclav Kotesovec, Nov 28 2017

D-finite with recurrence: 4782969*(9*n + 55)*(3*n + 5)*(3*n + 13)*(9*n - 1)*(9*n + 23)*(9*n + 47)*(9*n + 7)*(9*n + 31)*a(n) + 18600435*(2*n + 7)*(n + 1)*(2125764*n^6 + 44641044*n^5 + 367875270*n^4 + 1504568520*n^3 + 3174773616*n^2 + 3225344346*n + 1206357785)*a(n + 1) + 56260575*(n + 1)*(n + 2)*(2125764*n^6 + 51018336*n^5 + 496602090*n^4 + 2503677600*n^3 + 6877447236*n^2 + 9730412064*n + 5520476615)*a(n + 2) + 11814720750*(2*n + 9)*(n + 3)*(n + 2)*(n + 1)*(4374*n^4 + 78732*n^3 + 516456*n^2 + 1459458*n + 1498345)*a(n + 3) + 12762815625*(n + 1)*(n + 2)*(n + 3)*(n + 4)*(4374*n^4 + 87480*n^3 + 649296*n^2 + 2118960*n + 2564591)*a(n + 4) + 536038256250*(2*n + 11)*(n + 5)*(n + 4)*(n + 3)*(n + 2)*(n + 1)*(18*n^2 + 198*n + 535)*a(n + 5) + 77207156250*(n + 1)*(n + 2)*(n + 3)*(n + 4)*(n + 5)*(n + 6)*(54*n^2 + 648*n + 1939)*a(n + 6) + 257357187500*(2*n + 13)*(n + 7)*(n + 6)*(n + 5)*(n + 4)*(n + 3)*(n + 2)*(n + 1)*a(n + 7) - 26495939759*(n + 1)*(n + 2)*(n + 3)*(n + 4)*(n + 5)*(n + 6)*(n + 7)*(n + 8)*a(n + 8) = 0.

EXAMPLE

A(x) = 1 + 3*x + 12*x^2 + 180*x^3 + 3018*x^4 + 56238*x^5 +...

A(x)^9 = 1 + 27*x + 432*x^2 + 6480*x^3 + 108648*x^4 + 2024568*x^5 +...

MATHEMATICA

CoefficientList[1 + InverseSeries[Series[(1+36*x - (1+x)^9)/81, {x, 0, 20}], x], x] (* Vaclav Kotesovec, Nov 28 2017 *)

PROG

(PARI) {a(n)=local(A=1+3*x+12*x^2+x*O(x^n)); for(i=0, n, A=A+(-36*A+35+81*x+A^9)/27); polcoeff(A, n)}

CROSSREFS

Cf. A120588 - A120605, A120607.

Sequence in context: A004168 A301650 A061960 * A089428 A211892 A063801

Adjacent sequences: A120603 A120604 A120605 * A120607 A120608 A120609

KEYWORD