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

0,3

Links

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

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

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 *)

Prog

(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

Crossrefs

Cf. A253192.

Sequence in context: A122008 A248254 A212918 * A253192 A036254 A047170

Adjacent sequences: A255194 A255195 A255196 * A255198 A255199 A255200

Keyword

nonn,easy

Author

Michael D. Weiner, Feb 16 2015

Extensions

Definition clarified by Michael D. Weiner, Mar 09 2015

Status

approved