A295809 G.f. A(x) satisfies: A(x)^3 = 1+x + x*(A(x)^2 + A(x)^7).

1, 1, 2, 9, 44, 238, 1363, 8129, 49947, 313982, 2009804, 13054923, 85835763, 570162938, 3820449453, 25792692527, 175277931388, 1198017908942, 8230391249548, 56801549666858, 393622890736512, 2737841490099777, 19107124975145342, 133755761621788177, 938960164858527807, 6608463199584560132, 46621379983243723382, 329627663780846842009

Offset: 0

Author

Paul D. Hanna, Nov 27 2017

Keywords

• nonn

Comments

Compare to: G(x)^3 = 1+x + x*(G(x)^2 + G(x)^4) holds when G(x) = (1 + x - sqrt(1 - 2*x - 3*x^2))/(2*x) is a g.f. of the Motzkin numbers (A001006).

Examples

			G.f.: A(x) = 1 + x + 2*x^2 + 9*x^3 + 44*x^4 + 238*x^5 + 1363*x^6 + 8129*x^7 + 49947*x^8 + 313982*x^9 + 2009804*x^10 + 13054923*x^11 + 85835763*x^12 +...
such that A(x)^3 = 1+x + x*(A(x)^2 + A(x)^7).
RELATED SERIES.
A(x)^3 = 1 + 3*x + 9*x^2 + 40*x^3 + 204*x^4 + 1125*x^5 + 6536*x^6 + 39390*x^7 + 243966*x^8 + 1543350*x^9 + 9929589*x^10 +...
A(x)^2 = 1 + 2*x + 5*x^2 + 22*x^3 + 110*x^4 + 600*x^5 + 3459*x^6 + 20728*x^7 + 127824*x^8 + 805852*x^9 + 5170270*x^10 +...
A(x)^7 = 1 + 7*x + 35*x^2 + 182*x^3 + 1015*x^4 + 5936*x^5 + 35931*x^6 + 223238*x^7 + 1415526*x^8 + 9123737*x^9 + 59601227*x^10 +...
		

Links

• Paul D. Hanna, Table of n, a(n) for n = 0..520

Crossrefs

Cf. A295808, A295810. A259757, A295537, A295538.

Programs

PARI

{a(n) = my(A=1+x); for(i=1, n, A = ((1+x) + x*A^2 + x*A^7 +x*O(x^n))^(1/3) ); polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))

PARI

{a(n) = my(A=1+x); A = 1 + serreverse( x/(1 + 2*x + 5*x^2 + 6*x^3 + 4*x^4 + x^5 +x*O(x^n)) ); polcoeff(A,n)}
for(n=0, 30, print1(a(n), ", "))

Formula

G.f. A(x) satisfies:

(1) A(x) = 1 + Series_Reversion( x/(1 + 2*x + 5*x^2 + 6*x^3 + 4*x^4 + x^5) ).

(2) F(A(x)) = x such that F(x) = -(1-x)/(1 - x + x^2 - x^4 + x^5).