Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).
%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