OFFSET
0,3
COMMENTS
a(n) is the number of dissections of a regular (n+2)-gon using 0 or more strictly disjoint diagonals.
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..1000
Jean-Luc Baril, Sergey Kirgizov, Rémi Maréchal, and Vincent Vajnovszki, Grand Dyck paths with air pockets, arXiv:2211.04914 [math.CO], 2022.
Samuele Giraudo, Combalgebraic structures on decorated cliques, Formal Power Series and Algebraic Combinatorics, Séminaire Lotharingien de Combinatoire, 78B.15, 2017, p. 8; arXiv:1709.08416 [math.CO], 2017.
Giovanni Resta, Illustration of a(3)-a(10)
FORMULA
G.f.: 1 + (1+x)*( 1 -2*x -x^3 - sqrt((1 -3*x+ x^2)*(1-x)*(1-x^3)) )/(2*x^4).
Logarithmic derivative yields A132461. - Paul D. Hanna, Nov 09 2013
G.f.: exp( Sum_{n>=1} A132461(n)*x^n/n ), where A132461(n) = Sum_{k=0..[n/2]} (C(n-k,k) + C(n-k-1,k-1))^2. - Paul D. Hanna, Nov 09 2013
EXAMPLE
a(3)=6 because there are 5 ways to insert a single diagonal into a pentagon plus the empty dissection.
MAPLE
seq(coeff(series(1 + (1+x)*( 1 -2*x -x^3 - sqrt((1 -3*x+ x^2)*(1-x)*(1-x^3)) )/(2*x^4), x, n+2), x, n), n = 0..40); # G. C. Greubel, Dec 28 2019
MATHEMATICA
CoefficientList[Series[1 +(1+x)*(1-2*x-x^3 -Sqrt[(1-3*x+x^2)*(1-x)*(1-x^3)])/( 2*x^4), {x, 0, 40}], x] (* G. C. Greubel, Dec 28 2019 *)
PROG
(PARI) {A132461(n)=sum(k=0, n\2, (binomial(n-k, k)+binomial(n-k-1, k-1))^2)}
{a(n)=polcoeff(exp(sum(m=1, n, A132461(m)*x^m/m)+x*O(x^n)), n)} \\ Paul D. Hanna, Nov 09 2013
(Magma) R<x>:=PowerSeriesRing(Rationals(), 40); Coefficients(R!( 1 + (1+x)*( 1 -2*x -x^3 - Sqrt((1 -3*x+ x^2)*(1-x)*(1-x^3)) )/(2*x^4) )); // G. C. Greubel, Dec 28 2019
(Sage)
def A093128_list(prec):
P.<x> = PowerSeriesRing(ZZ, prec)
return P( 1 + (1+x)*( 1 -2*x -x^3 - sqrt((1 -3*x+ x^2)*(1-x)*(1-x^3)) )/(2*x^4) ).list()
A093128_list(40) # G. C. Greubel, Dec 28 2019
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
David Callan, Mar 23 2004
EXTENSIONS
Terms a(26) onward added by G. C. Greubel, Dec 28 2019
STATUS
approved