This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A217596 #63 Jul 29 2025 21:38:07
%S A217596 1,-1,-2,-5,-17,-64,-259,-1098,-4815,-21659,-99385,-463385,-2189070,
%T A217596 -10455340,-50402858,-244929608,-1198504743,-5900360016,-29204546125,
%U A217596 -145244328630,-725451444795,-3637422742470,-18301949731665,-92380935149100,-467659449093330
%N A217596 Expansion of g.f. x / reversion(x - x^2 - x^3).
%C A217596 For n>0, -a(n) is the number of times a convex polygon with n + 2 sides can be divided into triangles and quadrilaterals by non-intersecting diagonals such that none of the dividing diagonals passes through a chosen vertex. - _Muhammed Sefa Saydam_, Jan 27 2025
%D A217596 H. S. Wilf, Generatingfunctionology, Academic Press, NY, 1990.
%H A217596 G. C. Greubel, <a href="/A217596/b217596.txt">Table of n, a(n) for n = 0..600</a>
%F A217596 a(n) = -Sum_{i=ceiling(n/2)..n} binomial(i,n-i)*binomial(n+i-2,n-2)/(n-1), n>1, a(0)=1, a(1)=-1.
%F A217596 a(n) = -Sum_{i=0..n-1} A001002(n-i)*a(i), a(0)=1. [corrected by _Muhammed Sefa Saydam_, Jul 23 2025]
%F A217596 From _Paul D. Hanna_, Mar 19 2013: (Start)
%F A217596 G.f. satisfies:
%F A217596 (1) A(x) = 1 - x/A(x) - x^2/A(x)^2.
%F A217596 (2) A(x - x^2 - x^3) = 1 - x - x^2.
%F A217596 (3) [x^n] A(x)^n = -A000204(n), where A000204 is the Lucas numbers.
%F A217596 (4) [x^n] A(x)^(n+1) = 0 for n>2. (End)
%F A217596 Conjecture: 25*n*(n-1)*a(n) - 5*(n-1)*(25*n-42)*a(n-1) + 3*(-23*n^2 + 59*n + 4)*a(n-2) + 9*(3*n-10)*(3*n-11)*a(n-3)=0. - _R. J. Mathar_, May 23 2014
%F A217596 Maple's sumrecursion command gives the second-order recurrence equation: 5*n*(n - 1)*(4*n - 9)*a(n) = 2*(n - 1)*(44*n^2 - 165*n + 150)*a(n-1) + 3*(4*n - 5)(3*n - 7)(3*n - 8)*a(n-2) with initial conditions a(1) = -1 and a(2) = -2. Mathar's conjectured third-order recurrence follows from this. - _Peter Bala_, Feb 15 2015
%F A217596 a(n) = -A001002(n-1) - Sum_{i=0..n-2} A001002(i) * A001002(n-i-2) for n >= 2. - _Muhammed Sefa Saydam_, Jul 25 2025
%e A217596 G.f.: A(x) = 1 - x - 2*x^2 - 5*x^3 - 17*x^4 - 64*x^5 - 259*x^6 - 1098*x^7 - ...
%t A217596 CoefficientList[x/InverseSeries[Series[x-x^2-x^3, {x, 0, 20}], x],x] (* _Vaclav Kotesovec_, Feb 15 2015 *)
%o A217596 (Maxima)
%o A217596 a(n):=if n=0 then 1 else if n=1 then -1 else -sum(binomial(i,n-i)*binomial(n+i-2,n-2),i,ceiling(n/2),n)/(n-1);
%o A217596 (PARI) {a(n)=polcoeff(x/serreverse(x-x^2-x^3+x^2*O(x^n)),n)} \\ _Paul D. Hanna_, Mar 19 2013
%o A217596 (PARI) /* Using Vladimir Kruchinin's binomial sum: */
%o A217596 {a(n)=if(n==0,1,if(n==1,-1,-sum(i=n\2,n,binomial(i,n-i)*binomial(n+i-2,n-2))/(n-1)))} \\ _Paul D. Hanna_, Mar 19 2013
%o A217596 (PARI) {a(n)=local(A=1);for(i=1,n,A=1-x/A-x^2/A^2+x*O(x^n));polcoeff(A,n)} \\ _Paul D. Hanna_, Mar 19 2013
%Y A217596 Cf. A001002, A007440.
%K A217596 sign,easy
%O A217596 0,3
%A A217596 _Vladimir Kruchinin_, Mar 19 2013