A255197 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.

1, 0, 5, 6, 35, 80, 306, 880, 3003, 9384, 31070, 100226, 330015, 1079392, 3559001, 11724930, 38772445, 128313480, 425553513, 1412911148, 4697992880, 15637660896, 52109660575, 173809285676, 580261793715, 1938778221800, 6482844907190, 21692435752290, 72633495206803

Offset: 0

Author

Michael D. Weiner, Feb 16 2015

Keywords

• nonn
• easy

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000

Crossrefs

Cf. A253192.

Programs

Maple

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));

Mathematica

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 *)

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

Formula

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.

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

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

Extensions

Definition clarified by Michael D. Weiner, Mar 09 2015