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.

0, 1, 6, 31, 156, 785, 3978, 20335, 104856, 545073, 2854350, 15046383, 79787700, 425360481, 2278586898, 12259138975, 66216193968, 358941938849, 1952111592342, 10648449309823, 58245727453260, 319406931168241, 1755674399021466, 9671384910586511

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 and 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.: (1-3*z-sqrt(1-6*z+z^2))^2/(16*z^3).

a(n) = (1/Pi)*Integral_{x=3-2*sqrt(2)..3+2*sqrt(2)} x^n*sqrt(-x^2+6x-1)*(x-3)/8. - 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

From Peter Bala, Aug 30 2023: (Start)

a(n) = Sum_{k = 0..n-1} 2^(k+1)/(n+1) * binomial(n+1, k)*binomial(n+1, k+2).

(n+3)*(n-1)*a(n) = 3*n*(2*n+1)*a(n-1) - n*(n-1)*a(n-2) with a(0) = 0 and a(1) = 1.

G.f. A(x) satisfies the algebraic equation 4*x^3*A(x)^2 - (5*x^2 - 6*x + 1)*A(x) + x = 0 and the differential equation

(3*x^4 - 19*x^3 + 9*x^2 - x)*dA/dx + (3*x^3 - 29*x^2 + 21*x - 3)*A(x) + 4*x = 0 with A(0) = 0. (End)

Maple

a := proc(n) option remember; if n = 0 then 0 elif n = 1 then 1 else (3*n*(2*n+1)*a(n-1) - n*(n-1)*a(n-2))/((n+3)*(n-1)) end if; end:
seq(a(n), n = 0..20); # Peter Bala, Aug 30 2023

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 A334650

Adjacent sequences: A065093 A065094 A065095 * A065097 A065098 A065099

Keyword

nonn,easy

Author

Wouter Meeussen, Nov 11 2001

Status

approved