A120607 G.f. satisfies: 37*A(x) = 36 + 81*x + A(x)^10, starting with [1,3,15].

1, 3, 15, 270, 5505, 124818, 3028200, 76896180, 2018211930, 54311811330, 1490518569747, 41556060361920, 1173726329836125, 33513124885393020, 965755118941566180, 28051840723006217040, 820439774630057541690

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..663

Formula

G.f.: A(x) = 1 + Series_Reversion((1+37*x - (1+x)^10)/81). Lagrange Inversion yields: G.f.: A(x) = Sum_{n>=0} C(9*n,n)/(8*n+1) * (36+81*x)^(8*n+1)/37^(9*n+1). - Paul D. Hanna, Jan 24 2008

a(n) ~ 3^(-1 + 4*n) * (-36 + 9*(37/10)^(10/9))^(1/2 - n) / (2^(5/9) * 5^(1/18) * 37^(4/9) * n^(3/2) * sqrt(Pi)). - Vaclav Kotesovec, Nov 28 2017

D-finite with recurrence: -3826375200*(5*n + 22)*(2*n + 7)*(5*n + 13)*(10*n + 71)*(10*n + 17)*(5*n + 31)*(5*n + 4)*(10*n + 53)*(10*n - 1)*a(n) - 3401222400*(n + 1)*(56250000*n^8 + 1800000000*n^7 + 24268125000*n^6 + 179235000000*n^5 + 788667665625*n^4 + 2101882650000*n^3 + 3281727850875*n^2 + 2708138007000*n + 886610005456)*a(n + 1) - 566870400000*(2*n + 9)*(n + 2)*(n + 1)*(300000*n^6 + 8100000*n^5 + 87922500*n^4 + 489105000*n^3 + 1464716275*n^2 + 2231043975*n + 1346040822)*a(n + 2) - 1175731200000*(n + 1)*(n + 2)*(n + 3)*(300000*n^6 + 9000000*n^5 + 110587500*n^4 + 711750000*n^3 + 2528196975*n^2 + 4694469750*n + 3556081921)*a(n + 3) - 9797760000000*(2*n + 11)*(n + 4)*(n + 3)*(n + 2)*(n + 1)*(12000*n^4 + 264000*n^3 + 2137000*n^2 + 7535000*n + 9762793)*a(n + 4) - 1741824000000*(n + 1)*(n + 2)*(n + 3)*(n + 4)*(n + 5)*(60000*n^4 + 1440000*n^3 + 12867000*n^2 + 50724000*n + 74424793)*a(n + 5) - 387072000000000*(2*n + 13)*(n + 6)*(n + 5)*(n + 4)*(n + 3)*(n + 2)*(n + 1)*(40*n^2 + 520*n + 1669)*a(n + 6) - 49152000000000*(n + 1)*(n + 2)*(n + 3)*(n + 4)*(n + 5)*(n + 6)*(n + 7)*(120*n^2 + 1680*n + 5869)*a(n + 7) - 327680000000000*(2*n + 15)*(n + 8)*(n + 7)*(n + 6)*(n + 5)*(n + 4)*(n + 3)*(n + 2)*(n + 1)*a(n + 8) + 27001782375529*(n + 1)*(n + 2)*(n + 3)*(n + 4)*(n + 5)*(n + 6)*(n + 7)*(n + 8)*(n + 9)*a(n + 9) = 0. - Robert Israel, Mar 20 2026

Example

A(x) = 1 + 3*x + 15*x^2 + 270*x^3 + 5505*x^4 + 124818*x^5 +...
A(x)^10 = 1 + 30*x + 555*x^2 + 9990*x^3 + 203685*x^4 + 4618266*x^5 +...

Mathematica

CoefficientList[1 + InverseSeries[Series[(1+37*x - (1+x)^10)/81, {x, 0, 20}], x], x] (* Vaclav Kotesovec, Nov 28 2017 *)

Prog

(PARI) {a(n)=local(A=1+3*x+15*x^2+x*O(x^n)); for(i=0, n, A=A+(-37*A+36+81*x+A^10)/27); polcoeff(A, n)}

Crossrefs

Cf. A120588 - A120606.

Sequence in context: A374342 A122591 A326263 * A013352 A013354 A013356

Adjacent sequences: A120604 A120605 A120606 * A120608 A120609 A120610

Keyword

nonn

Author

Paul D. Hanna, Jun 16 2006

Status

approved