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A065096 Sums of lists produced by a variant of the iteration that produces the Catalan numbers: start with 0 and at each iteration replace each integer k with the list 0,1,...,k-1,k,k+1,k,k-1,...,1,0 and let a(n) be the sum of the resulting (flattened) list after n iterations. 7
0, 1, 6, 31, 156, 785, 3978, 20335, 104856, 545073, 2854350, 15046383, 79787700, 425360481, 2278586898, 12259138975, 66216193968, 358941938849, 1952111592342, 10648449309823, 58245727453260, 319406931168241, 1755674399021466, 9671384910586511 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

Number of diagonals emanating from a fixed vertex of a convex (n+3)-gon in all of its dissections. Example: a(1)=1 because in the three dissections of a convex quadrilateral ABCD (namely: empty, {AC}, {BD}) there is only one diagonal emanating from A.

LINKS

Fung Lam, Table of n, a(n) for n = 0..1000

S. B. Ekhad, M. Yang, Proofs of Linear Recurrences of Coefficients of Certain Algebraic Formal Power Series Conjectured in the On-Line Encyclopedia Of Integer Sequences, (2017)

FORMULA

G.f.=(1-3z-sqrt(1-6z+z^2))^2/(16z^3).

a(n)=(1/pi)*Int(x^n*sqrt(-x^2+6x-1)*(x-3)/8,x,3-2sqrt(2),3+2sqrt(2)); - Paul Barry, Sep 16 2006

a(0) = 0 and, for n>0, a(n) = Sum_{k=1..n} A001003(k)*A001003(n+1-k) . - Philippe Deléham, Jan 27 2004

Conjecture: (n+3)*a(n) +3*(-3*n-4)*a(n-1) +(19*n-9)*a(n-2) +3*(-n+2)*a(n-3)=0. - R. J. Mathar, Nov 24 2012

Recurrence: (n+3)a(n) = -9*(n-3)*a(n-4) + 30*(2*n-3)*a(n-3) - 46*n*a(n-2) + 6*(2*n+3)*a(n-1). - Fung Lam, Jan 29 2014

a(n) ~ (3*sqrt(2)-4)^(3/2) * (3+2*sqrt(2))^(n+3) / (4 * sqrt(Pi) * n^(3/2)). - Vaclav Kotesovec, Feb 13 2014

MATHEMATICA

Table[Plus@@Flatten[Nest[ #/.a_Integer:> Join[Range[0, a+1], Range[a, 0, -1]]&, {0}, n]], {n, 0, 10}]

Table[Range[n, 0, -1].Table[a[n, k], {k, 0, n}], {n, 0, 36}] with a[n, k] as defined in A033877.

CROSSREFS

Cf. A000108, A001003.

Sequence in context: A003463 A026771 A289788 * A077352 A038223 A022034

Adjacent sequences:  A065093 A065094 A065095 * A065097 A065098 A065099

KEYWORD

nonn

AUTHOR

Wouter Meeussen, Nov 11 2001

STATUS

approved

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Last modified October 16 19:22 EDT 2018. Contains 316271 sequences. (Running on oeis4.)