cp's OEIS Frontend

A253194 Number of ways to place non-intersecting diagonals in a convex (n+2)-gon so as to create no pentagons.

%I A253194 #22 Apr 07 2015 12:53:12
%S A253194 1,3,10,39,162,707,3190,14766,69719,334481,1625846,7989908,39631204,
%T A253194 198151579,997623275,5053274850,25734158411,131680565544,676693557574,
%U A253194 3490897656337,18071699948492,93851485181749,488815126122166
%N A253194 Number of ways to place non-intersecting diagonals in a convex (n+2)-gon so as to create no pentagons.
%H A253194 D. Birmajer, J. B. Gil, and M. Weiner, <a href="http://arxiv.org/abs/1503.05242">Colored partitions of a convex polygon by noncrossing diagonals</a>, arXiv:1503.05242 [math.CO], 2015.
%F A253194 a(n) = (1/(n+1))*Sum_{i=0..floor(n/3)} Sum_{k=i+1..n-2*i} (-1)^i*binomial(n+k,k)*binomial(k,i)*binomial(n-3*i-1,k-i-1), n !== 0 (mod 3),
%F A253194 a(n) = ((-1)^(n/3)/(n+1))*binomial(4*n/3,n/3) + (1/(n+1))*Sum_{i=0..(n/3)-1} Sum_{k=i+1..n-2*i} (-1)^i*binomial(n+k,k)*binomial(k,i)*binomial(n-3*i-1,k-i-1), n == 0 (mod 3).
%F A253194 Recurrence: 275*(n-2)*(n-1)*n*(n+1)*(13962464*n^5 - 196202616*n^4 + 1069508732*n^3 - 2802358002*n^2 + 3480787751*n - 1597000860)*a(n) = 900*(n-2)*(n-1)*n*(27924928*n^6 - 406367696*n^5 + 2336399896*n^4 - 6678345644*n^3 + 9735406192*n^2 - 6526643891*n + 1424056473)*a(n-1) - 8*(n-2)*(n-1)*(1870970176*n^7 - 30033090896*n^6 + 197840216728*n^5 - 682911269612*n^4 + 1297104157966*n^3 - 1273486799084*n^2 + 486871358313*n + 21712608900)*a(n-2) - 16*(n-2)*(2569093376*n^8 - 47662201536*n^7 + 375012676176*n^6 - 1627682459628*n^5 + 4239503473896*n^4 - 6734585440155*n^3 + 6299789310412*n^2 - 3112752665481*n + 598681926090)*a(n-3) + 16*(2457393664*n^9 - 54190809728*n^8 + 518749193184*n^7 - 2816319789288*n^6 + 9492888047100*n^5 - 20388222826734*n^4 + 27407291375141*n^3 - 21462176121217*n^2 + 8117426803296*n - 745665750648)*a(n-4) - 8*(n-4)*(2*n - 7)*(4*n - 17)*(4*n - 11)*(13962464*n^5 - 126390296*n^4 + 424322908*n^3 - 631422862*n^2 + 369599799*n - 31302531)*a(n-5). - _Vaclav Kotesovec_, Mar 30 2015
%e A253194 a(3)=10 because the pentagon allows all but the null placement, i.e., 5 placements of 1 diagonal and 5 placements of two diagonals.
%t A253194 Rest[CoefficientList[(InverseSeries[Series[(y-2*y^2+y^4-y^5)/(1-y),{y,0,24}],x]-x)/x,x]]
%o A253194 (PARI) A253194(n)=sum(i=0,(n-1)\3,sum(k=i+1,n-2*i, (-1)^i*binomial(n+k,k)*binomial(k,i)*binomial(n-3*i-1,k-i-1)),if(n%3==0,(-1)^(n/3)*binomial(4*n/3,n/3)))/(n+1) \\ _M. F. Hasler_, Apr 07 2015
%Y A253194 Cf. A046736, A054515.
%K A253194 nonn
%O A253194 1,2
%A A253194 _Michael D. Weiner_, Mar 24 2015