A255197 - OEIS

%I #40 Apr 08 2017 05:19:37

%S 1,0,5,6,35,80,306,880,3003,9384,31070,100226,330015,1079392,3559001,

%T 11724930,38772445,128313480,425553513,1412911148,4697992880,

%U 15637660896,52109660575,173809285676,580261793715,1938778221800,6482844907190,21692435752290,72633495206803

%N Number of dissections of a convex polygon with n+3 sides that have exactly one triangle, and that triangle shares at least one side with the exterior polygon.

%H G. C. Greubel, <a href="/A255197/b255197.txt">Table of n, a(n) for n = 0..1000</a>

%F a(n) = (n+3)/(n+2) * Sum_{k=1..n/2} C(n+k+1,k)*C(n-k-1,k-1)*(n+2*k)/(n+k+1) , n>0.

%F Recurrence: 5*(n-2)*n*(n+1)^2*(n+2)^2*(481*n^2 - 1003*n + 540)*a(n) = 4*n*(n+1)^2*(n+3)*(962*n^4 - 3449*n^3 + 3262*n^2 - 784*n - 180)*a(n-1) + 4*(n-1)*n*(n+2)*(n+3)*(3848*n^4 - 11872*n^3 + 12073*n^2 - 3572*n - 180)*a(n-2) - 2*(n-2)*(n-1)*(n+1)*(n+2)*(n+3)*(2*n-5)*(481*n^2 - 41*n + 18)*a(n-3). - _Vaclav Kotesovec_, Feb 19 2015

%F a(n) ~ c * d^n / sqrt(Pi*n), where d = 3.40869819984215108586... is the root of the equation 4 - 32*d - 8*d^2 + 5*d^3 = 0, and c = 0.838651324525827608604668464... is the root of the equation 169 + 157184*c^2 - 275872*c^4 + 74000*c^6 = 0. - _Vaclav Kotesovec_, Feb 21 2015

%p a:=n->(n+3)/(n+2)*sum(binomial(n+k+1,k)*binomial(n-k-1,k-1)*(n+2*k)/(n+k+1),k=1..trunc(n/2)): (1,seq(a(n), n=1..30));

%t Flatten[{1,Table[(n+3)/(n+2)*Sum[Binomial[n+k+1,k]*Binomial[n-k-1,k-1]*(n+2k)/(n+k+1),{k,Floor[n/2]}],{n,20}]}] (* _Vaclav Kotesovec_, Feb 19 2015 *)

%o (PARI) a(n) = if (n==0, 1, (n+3)*sum(k=1, n\2, binomial(n+k+1,k)*binomial(n-k-1,k-1)*(n+2*k)/(n+k+1))/(n+2)); \\ _Michel Marcus_, Mar 03 2015

%Y Cf. A253192.

%K nonn,easy

%O 0,3

%A _Michael D. Weiner_, Feb 16 2015

%E Definition clarified by _Michael D. Weiner_, Mar 09 2015