%I #32 Nov 14 2022 20:02:41

%S 1,1,3,6,13,29,65,148,341,793,1860,4395,10452,24999,60097,145130,

%T 351916,856502,2091599,5123437,12585354,30995031,76516348,189310421,

%U 469335998,1165790119,2900870597,7230320746,18049387617,45123390441,112963369113,283162526640,710664478791,1785645155847,4491596869206

%N Number of dissections of a polygon using strictly disjoint diagonals.

%C a(n) is the number of dissections of a regular (n+2)-gon using 0 or more strictly disjoint diagonals.

%H G. C. Greubel, <a href="/A093128/b093128.txt">Table of n, a(n) for n = 0..1000</a>

%H Jean-Luc Baril, Sergey Kirgizov, Rémi Maréchal, and Vincent Vajnovszki, <a href="https://arxiv.org/abs/2211.04914">Grand Dyck paths with air pockets</a>, arXiv:2211.04914 [math.CO], 2022.

%H Samuele Giraudo, <a href="https://arxiv.org/abs/1709.08416">Combalgebraic structures on decorated cliques</a>, Formal Power Series and Algebraic Combinatorics, Séminaire Lotharingien de Combinatoire, 78B.15, 2017, p. 8; arXiv:1709.08416 [math.CO], 2017.

%H Giovanni Resta, <a href="/A093128/a093128.pdf">Illustration of a(3)-a(10)</a>

%F G.f.: 1 + (1+x)*( 1 -2*x -x^3 - sqrt((1 -3*x+ x^2)*(1-x)*(1-x^3)) )/(2*x^4).

%F a(n) = A004148(n+2) - A004148(n) for n>=1.

%F Logarithmic derivative yields A132461. - _Paul D. Hanna_, Nov 09 2013

%F 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

%e a(3)=6 because there are 5 ways to insert a single diagonal into a pentagon plus the empty dissection.

%p 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

%t 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 *)

%o (PARI) {A132461(n)=sum(k=0,n\2,(binomial(n-k, k)+binomial(n-k-1, k-1))^2)}

%o {a(n)=polcoeff(exp(sum(m=1,n,A132461(m)*x^m/m)+x*O(x^n)),n)} \\ _Paul D. Hanna_, Nov 09 2013

%o (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

%o (Sage)

%o def A093128_list(prec):

%o P.<x> = PowerSeriesRing(ZZ, prec)

%o 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()

%o A093128_list(40) # _G. C. Greubel_, Dec 28 2019

%Y Row sums of A093127.

%K easy,nonn

%O 0,3

%A _David Callan_, Mar 23 2004

%E Terms a(26) onward added by _G. C. Greubel_, Dec 28 2019