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A093128 Number of dissections of a polygon using strictly disjoint diagonals. 3
1, 1, 3, 6, 13, 29, 65, 148, 341, 793, 1860, 4395, 10452, 24999, 60097, 145130, 351916, 856502, 2091599, 5123437, 12585354, 30995031, 76516348, 189310421, 469335998, 1165790119, 2900870597, 7230320746, 18049387617, 45123390441, 112963369113, 283162526640, 710664478791, 1785645155847, 4491596869206 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

a(n) is the number of dissections of a regular (n+2)-gon using 0 or more strictly disjoint diagonals.

LINKS

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

Samuele Giraudo, Combalgebraic structures on decorated cliques, Formal Power Series and Algebraic Combinatorics, Séminaire Lotharingien de Combinatoire, 78B.15, 2017, p. 8; arXiv:1709.08416 [math.CO], 2017.

Giovanni Resta, Illustration of a(3)-a(10)

FORMULA

G.f.: 1 + (1+x)*( 1 -2*x -x^3 - sqrt((1 -3*x+ x^2)*(1-x)*(1-x^3)) )/(2*x^4).

a(n) = A004148(n+2) - A004148(n) for n>=1.

Logarithmic derivative yields A132461. - Paul D. Hanna, Nov 09 2013

G.f.: exp( Sum_{n>=1} A132461(n)*x^n/n ), where A132461(n) = Sum_{k=0..[n/2]} (C(n-k,k) + C(n-k-1,k-1))^2. - Paul D. Hanna, Nov 09 2013

EXAMPLE

a(3)=6 because there are 5 ways to insert a single diagonal into a pentagon plus the empty dissection.

MAPLE

seq(coeff(series(1 + (1+x)*( 1 -2*x -x^3 - sqrt((1 -3*x+ x^2)*(1-x)*(1-x^3)) )/(2*x^4), x, n+2), x, n), n = 0..40); # G. C. Greubel, Dec 28 2019

MATHEMATICA

CoefficientList[Series[1 +(1+x)*(1-2*x-x^3 -Sqrt[(1-3*x+x^2)*(1-x)*(1-x^3)])/( 2*x^4), {x, 0, 40}], x] (* G. C. Greubel, Dec 28 2019 *)

PROG

(PARI) {A132461(n)=sum(k=0, n\2, (binomial(n-k, k)+binomial(n-k-1, k-1))^2)}

{a(n)=polcoeff(exp(sum(m=1, n, A132461(m)*x^m/m)+x*O(x^n)), n)} \\ Paul D. Hanna, Nov 09 2013

(MAGMA) R<x>:=PowerSeriesRing(Rationals(), 40); Coefficients(R!( 1 + (1+x)*( 1 -2*x -x^3 - Sqrt((1 -3*x+ x^2)*(1-x)*(1-x^3)) )/(2*x^4) )); // G. C. Greubel, Dec 28 2019

(Sage)

def A093128_list(prec):

    P.<x> = PowerSeriesRing(ZZ, prec)

    return P( 1 + (1+x)*( 1 -2*x -x^3 - sqrt((1 -3*x+ x^2)*(1-x)*(1-x^3)) )/(2*x^4) ).list()

A093128_list(40) # G. C. Greubel, Dec 28 2019

CROSSREFS

Row sums of A093127.

Sequence in context: A214260 A078061 A018909 * A005313 A213674 A108639

Adjacent sequences:  A093125 A093126 A093127 * A093129 A093130 A093131

KEYWORD

easy,nonn

AUTHOR

David Callan, Mar 23 2004

EXTENSIONS

Terms a(26) onward added by G. C. Greubel, Dec 28 2019

STATUS

approved

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Last modified January 15 18:16 EST 2021. Contains 340188 sequences. (Running on oeis4.)