A199670 -id:A199670

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A183611

E.g.f. satisfies: A'(x) = A(x)^2 + x*A(x)^3, with A(0) = 1.

+10
2

1, 1, 3, 14, 91, 756, 7657, 91504, 1260441, 19663280, 342669691, 6597811584, 139094618467, 3186675803584, 78834061767825, 2094418664339456, 59474007876381553, 1797637447068293376, 57623116235327599411

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,3

LINKS

Vaclav Kotesovec, Table of n, a(n) for n = 0..400

V. Dotsenko, Pattern avoidance in labelled trees, arXiv preprint arXiv:1110.0844 [math.CO], 2011-2012.

FORMULA

E.g.f.: A(x) = 1 + A(x)*[Integral 1 + x*A(x) dx], where the integration does not include the constant term.

E.g.f.: d/dx Series_Reversion(Sum_{n>=1} x^(3*n-2)/(3*n-2)! - x^(3*n-1)/(3*n-1)!).

a(n) ~ n^n * exp(Pi*(n+1)/(3*sqrt(3))-n). - Vaclav Kotesovec, Feb 19 2014

EXAMPLE

E.g.f.: A(x) = 1 + x + 3*x^2/2! + 14*x^3/3! + 91*x^4/4! +...

A'(x) = 1 + 3*x + 14*x^2/2! + 91*x^3/3! + 756*x^4/4! +...

A(x)^2 = 1 + 2*x + 8*x^2/2! + 46*x^3/3! + 348*x^4/4! + 3262*x^5/5! +...

A(x)^3 = 1 + 3*x + 15*x^2/2! + 102*x^3/3! + 879*x^4/4! + 4395*x^5/5! +...

E.g.f. A(x) = d/dx Series_Reversion(G(x)) where G(x) begins:

G(x) = x - x^2/2! + x^4/4! - x^5/5! + x^7/7! - x^8/8! + x^10/10! - x^11/11! +...

The series reversion of G(x) begins:

x + x^2/2! + 3*x^3/3! + 14*x^4/4! + 91*x^5/5! + 756*x^6/6! +...

MATHEMATICA

terms = 20; A[_] = 0;

Do[A[x_] = 1+Integrate[A[x]^2 + x A[x]^3, x]+O[x]^terms // Normal, terms];

CoefficientList[A[x], x] Range[0, terms-1]! (* Jean-François Alcover, Oct 27 2018 *)

PROG

(PARI) {a(n)=local(A=1); for(n=0, n, A=1+A*intformal(1+x*A+x*O(x^n))); n!*polcoeff(A, n)}

(PARI) {a(n)=n!*polcoeff(deriv(serreverse(sum(m=1, n\3+1, x^(3*m-2)/(3*m-2)!-x^(3*m-1)/(3*m-1)!+x^2*O(x^n)))), n)}

CROSSREFS

Cf. A199670, A049774.

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Mar 21 2011

STATUS

approved

A199675

Expansion of e.g.f. 1/(exp(-x) - Sum_{n>=0} (-x)^(3*n+2)/(3*n+2)!).

+10
1

1, 1, 2, 7, 31, 170, 1129, 8737, 77198, 767683, 8482519, 103093958, 1366897597, 19633740673, 303706037546, 5033465370031, 88983532209967, 1671402633292562, 33241154368669921, 697834148797749601, 15420722865332961206, 357805114894717632331, 8697446048869287663271

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,3

LINKS

Table of n, a(n) for n=0..22.

FORMULA

E.g.f.: A(x) = 1/Q(0); Q(k) = 1-x/((3*k+1)-(x^2)*(3*k+1)/((x^2)+3*(3*k+2)*(k+1)/Q(k+1))); (continued fraction). - Sergei N. Gladkovskii, Nov 26 2011

EXAMPLE

E.g.f.: A(x) = 1 + x + 2*x^2/2! + 7*x^3/3! + 31*x^4/4! + 170*x^5/5! +...

where

A(x) = 1/(1 - x - x^3/3! + x^4/4! + x^6/6! - x^7/7! - x^9/9! + x^10/10! +...).

PROG

(PARI) {a(n)=n!*polcoeff(1/(exp(-x+x*O(x^n)) - sum(m=0, n\3, (-x)^(3*m+2)/(3*m+2)! )), n)}

(PARI) {a(n)=n!*polcoeff(1/(sum(m=0, n\3+1, (-x)^(3*m)/(3*m)! + (-x)^(3*m+1)/(3*m+1)! +x^2*O(x^n))), n)}