OFFSET
0,2
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..900
Elżbieta Liszewska, Wojciech Młotkowski, Some relatives of the Catalan sequence, arXiv:1907.10725 [math.CO], 2019.
FORMULA
G.f. A(x) satisfies:
(1) A(x) = exp( x*(A(x) + A(x)^2) + Integral(A(x) + A(x)^2 dx) ).
(2) A(x) = (1/x)*Series_Reversion( x*(1-2*x-x^2)/(1+x)^2 ).
(3) A(x) = 1 + x*A(x)*(1 + A(x))*(2 + x*A(x)).
(4) A(x) = 1 + Sum_{n>=2} (-1)^n * 2*n * x^(n-1) * A(x)^n.
Recurrence: 2*n*(n+1)*(29*n-39)*a(n) = n*(754*n^2 - 1391*n + 519)*a(n-1) - (203*n^3 - 679*n^2 + 660*n - 180)*a(n-2) + (n-3)*(2*n-3)*(29*n-10)*a(n-3). - Vaclav Kotesovec, Dec 21 2013
a(n) ~ 1/174*sqrt(87)*sqrt((8946558 + 86826*sqrt(87))^(1/3)*((8946558 + 86826*sqrt(87))^(2/3) + 42978 + 174*(8946558 + 86826*sqrt(87))^(1/3))) / ((8946558 + 86826*sqrt(87))^(1/3) * sqrt(Pi)) * 6^(-n)*((16046 + 174*sqrt(87))^(2/3) + 634 + 26*(16046 + 174*sqrt(87))^(1/3))^n*(16046 + 174*sqrt(87))^(-n/3) * (1/n)^(3/2). - Vaclav Kotesovec, Dec 21 2013
EXAMPLE
G.f.: A(x) = 1 + 4*x + 26*x^2 + 208*x^3 + 1858*x^4 + 17764*x^5 +...
Related expansions.
(1 + x*A(x))^2 = 1 + 2*x + 9*x^2 + 60*x^3 + 484*x^4 + 4340*x^5 +...
(1 + A(x))/2 = 1 + 2*x + 13*x^2 + 104*x^3 + 929*x^4 + 8882*x^5 +...
A(x) + A(x)^2 = 2 + 12*x + 94*x^2 + 832*x^3 + 7914*x^4 + 78972*x^5 +...
log(A(x)) = 4*x + 36*x^2/2 + 376*x^3/3 + 4160*x^4/4 + 47484*x^5/5 +...
MATHEMATICA
CoefficientList[InverseSeries[Series[x*(1-2*x-x^2)/(1+x)^2, {x, 0, 20}], x]/x, x] (* Vaclav Kotesovec, Dec 21 2013 *)
PROG
(PARI) {a(n)=polcoeff((serreverse(x*(1-2*x-x^2)/(1+x)^2 +x^2*O(x^n))/x), n)}
for(n=0, 30, print1(a(n), ", "))
(PARI) {a(n)=local(A=1); for(i=1, n, A=exp(x*(A+A^2)+intformal(A+A^2 +x*O(x^n)))); polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))
(PARI) {a(n)=local(A=1); for(i=1, n, A=1+x*A*(1+A)*(2+x*A) +x*O(x^n)); polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Nov 10 2013
STATUS
approved