A275755 G.f. satisfies: A(x) = x + A( A(x)^2 - A(x)^5 ).

1, 1, 2, 6, 19, 65, 234, 873, 3346, 13099, 52154, 210541, 859768, 3545263, 14741148, 61736903, 260192880, 1102704585, 4696416190, 20090502706, 86285786519, 371917832707, 1608317086940, 6975728777332, 30338392601498, 132277349730004, 578075052215714, 2531710609461484, 11109852467209553, 48843541287179595, 215108137824940916, 948874606956945665, 4191979050580762418, 18545890698661636784, 82159569800859439840, 364432560308538162214, 1618431087549954575022

Offset: 1

Author

Paul D. Hanna, Aug 20 2016

Keywords

• nonn

Comments

Compare to: C(x) = x + C( C(x)^2 - C(x)^4 ) holds when C(x) = x + C(x)^2 is a g.f. of the Catalan numbers (A000108).

Examples

			G.f.: A(x) = x + x^2 + 2*x^3 + 6*x^4 + 19*x^5 + 65*x^6 + 234*x^7 + 873*x^8 + 3346*x^9 + 13099*x^10 + 52154*x^11 + 210541*x^12 + 859768*x^13 + 3545263*x^14 +...
such that A(x) = x + A( A(x)^2 - A(x)^5 ).
RELATED SERIES.
A(x)^2 = x^2 + 2*x^3 + 5*x^4 + 16*x^5 + 54*x^6 + 192*x^7 + 710*x^8 + 2702*x^9 + 10515*x^10 + 41660*x^11 + 167483*x^12 + 681532*x^13 + 2801816*x^14 +...
A(x)^5 = x^5 + 5*x^6 + 20*x^7 + 80*x^8 + 320*x^9 + 1286*x^10 + 5210*x^11 + 21285*x^12 + 87655*x^13 + 363660*x^14 + 1518952*x^15 +...
A(x^2 - x^5) = x^2 + x^4 - x^5 + 2*x^6 - 2*x^7 + 6*x^8 - 6*x^9 + 20*x^10 - 24*x^11 + 71*x^12 - 95*x^13 + 270*x^14 - 392*x^15 + 1063*x^16 - 1662*x^17 +...
where Series_Reversion(A(x)) = x - A(x^2 - x^5).
		

Links

• Paul D. Hanna, Table of n, a(n) for n = 1..300

Crossrefs

Cf. A275756, A275757, A275758, A141200.

Programs

PARI

{a(n) = my(A=x); for(i=1,n, A = x + subst(A,x, A^2 - A^5 +x*O(x^n))); polcoeff(A,n)}
for(n=1,40,print1(a(n),", "))

Formula

G.f. satisfies:

(1) A(x - A(x^2 - x^5)) = x.

(2) A(x) = x + Sum_{n>=0} d^n/dx^n A(x^2-x^5)^(n+1) / (n+1)!.

(3) A(x) = x * exp( Sum_{n>=0} d^n/dx^n A(x^2-x^5)^(n+1)/x / (n+1)! ).