0,3

a(n)= # of ways to dissect a convex (2n+2)-gon with non-crossing diagonals so that no (2m+1)-gons (m>0) appear - Len Smiley (smiley(AT)math.uaa.alaska.edu)

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

L. Carlitz, Enumeration of two-line arrays, Fib. Quart., 11 (1973), 113-130.

R. C. Read, On the enumeration of a class of plane multigraphs, Aequat. Math., 31 (1986), 47-63.

T. D. Noe, Table of n, a(n) for n=0..100

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 415

L. Smiley, Even-gon reference

Sum_{k=1..n} binomial(n, k)*binomial(2n+k, k-1)/n.

A(x)=Sum_{n>=0}a(n)x^(2n+1) satisfies (A-2*A^3)/(1-A^2)=x - Len Smiley (smiley(AT)math.uaa.alaska.edu).

4*n*(2*n + 1)*(17*n - 22)*a(n) = (1207*n^3 - 2769*n^2 + 1850*n - 360)*a(n - 1) - 2*(17*n - 5)*(n - 2)*(2*n - 3)*a(n - 2). - Vladeta Jovovic (vladeta(AT)eunet.rs), Jul 16 2004

G.f.: A(x) = 1/(1-G003169(x)) where G003169(x) is the g.f. of A003169. - Paul D. Hanna (pauldhanna(AT)juno.com), Nov 17 2004

a(2)=4 because we may place exactly one diagonal in 3 ways (forming 2 quadrilaterals), or not place any (leaving 1 hexagon).

Order := 40; solve(series((A-2*A^3)/(1-A^2), A)=x, A);

(PARI) a(n)=if(n<0, 0, polcoeff(serreverse((x-2*x^3)/(1-x^2)+O(x^(2*n+2))), 2*n+1))

(PARI) {a(n)=local(A=1+x+x*O(x^n)); for(i=1, n, A=(1+x*A)/(1-x*A)^2); sum(k=0, n, polcoeff(A^(n-k), k))} (Hanna)

Cf. A049124 (no 2m-gons).

Cf. A003169, A100327.

Sequence in context: A153291 A093965 A162480 this_sequence A032326 A007345 A099250

Adjacent sequences: A003165 A003166 A003167 this_sequence A003169 A003170 A003171

nonn,easy,nice

N. J. A. Sloane (njas(AT)research.att.com).