A068875 Expansion of (1 + x*C)*C, where C = (1 - (1 - 4*x)^(1/2))/(2*x) is the g.f. for Catalan numbers, A000108.

1, 2, 4, 10, 28, 84, 264, 858, 2860, 9724, 33592, 117572, 416024, 1485800, 5348880, 19389690, 70715340, 259289580, 955277400, 3534526380, 13128240840, 48932534040, 182965127280, 686119227300, 2579808294648, 9723892802904, 36734706144304, 139067101832008

Offset: 0

Author

N. J. A. Sloane, Jun 06 2002

Keywords

• nonn
• easy

Comments

A Catalan transform of A040000 under the mapping g(x) -> g(x*c(x)), where c(x) is the g.f. of A000108. Sequence A040000 can be retrieved using the mapping g(x) -> g(x*(1-x)). A040000(n) = Sum_{k=0..floor(n/2)} binomial(n-k, k) * (-1)^k * a(n-k). a(n) and A040000 may be described as a Catalan pair. - Paul Barry, Nov 14 2004

a(n) is the number of Dyck (n+1)-paths all of whose nonterminal descents to ground level are of odd length. For example, a(2) counts UUUDDD, UUDUDD, UDUUDD, UDUDUD. - David Callan, Jul 25 2005

From Gary W. Adamson, Jul 11 2011: (Start)

a(n) is the sum of the top row terms in M^n, where M is the following infinite square production matrix:

1, 1, 0, 0, 0, 0, ...

0, 1, 1, 0, 0, 0, ...

1, 1, 1, 1, 0, 0, ...

0, 1, 1, 1, 1, 0, ...

1, 1, 1, 1, 1, 1, ...

...

For example, the top row of M^3 = (2, 4, 3, 1) with sum = 10 = a(3). (End)

For n >= 1, a(n) is the number of binary trees with n+1 internal node in which one of the subtrees of the root is empty. Cf. A002057. [Sedgewick and Flajolet] - Geoffrey Critzer, Jan 05 2013

Empirical: a(n) is the number of entries of absolute value 1 that appear among all partitions in the canonical basis of the Temperley-Lieb algebra of order n. - John M. Campbell, Oct 17 2017

For n >= 1, a(n) is the number of Dyck paths of size n+2, whose corresponding unit interval graph has P3-hull number equal to 2. This result is due to Alrik Sandberg. - Per W. Alexandersson, Jan 09 2024

Examples

			G.f. = 1 + 2*x + 4*x^2 + 10*x^3 + 28*x^4 + 84*x^5 + 264*x^6 + 858*x^7 + ...
For example, the canonical basis of the Temperley-Lieb algebra of order 3 is {{{-3, 1}, {-2, -1}, {2, 3}}, {{-3, 3}, {-2, 2}, {-1, 1}}, {{-3, 3}, {-2, -1}, {1, 2}}, {{-3, -2}, {-1, 1}, {2, 3}}, {{-3, -2}, {-1, 3}, {1, 2}}}, and we see that the total number of entries of absolute value 1 that appear among the partitions in this basis is a(3) = 10.
		

References

• R. Sedgewick and P. Flajolet, Analysis of Algorithms, Addison Wesley, 1996, p. 225.

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Paul Barry, A Catalan Transform and Related Transformations on Integer Sequences, Journal of Integer Sequences, Vol. 8 (2005), Article 05.4.5.
• Paul Barry, Riordan Pseudo-Involutions, Continued Fractions and Somos 4 Sequences, arXiv:1807.05794 [math.CO], 2018.
• Alexander Burstein and Louis W. Shapiro, Pseudo-involutions in the Riordan group, arXiv:2112.11595 [math.CO], 2021.
• S. J. Cyvin, J. Brunvoll, E. Brendsdal, B. N. Cyvin and E. K. Lloyd, Enumeration of polyene hydrocarbons: a complete mathematical solution, J. Chem. Inf. Comput. Sci., 35 (1995), 743-751. [Annotated scanned copy]
• 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.
• Guo-Niu Han, Enumeration of Standard Puzzles, 2011. [Cached copy]
• Guo-Niu Han, Enumeration of Standard Puzzles, arXiv:2006.14070 [math.CO], 2020.

Crossrefs

A002420 and A262543 are essentially the same sequence as this.

Cf. A000108, A000984, A002057, A040000, A068875, A071721.

Programs

Magma

[1] cat [2*Binomial( 2*n, n)/(n+1): n in [1..30]]; // Vincenzo Librandi, Oct 17 2017

Maple

Z:=(1-sqrt(1-4*x))/2/x: G:=(2-(1+x)*Z)/Z: Gser:=series(-G, x=0, 30): (1,seq(coeff(Gser, x, n), n=2..26)); # Zerinvary Lajos, Dec 23 2006
Z:=-(1-z-sqrt(1-z))/sqrt(1-z): Zser:=series(Z, z=0, 32): (1,seq(coeff(Zser*4^n, z, n), n=2..26)); # Zerinvary Lajos, Jan 01 2007
A068875List := proc(m) local A, P, n; A := [1, 2]; P := [2];
for n from 1 to m - 2 do P := ListTools:-PartialSums([op(P), P[-1]]);
A := [op(A), P[-1]] od; A end: A068875List(26); # Peter Luschny, Mar 24 2022

Mathematica

nn=30;t=(1-(1-4x )^(1/2))/(2x);Prepend[Table[Coefficient[Series[1+x (y t -y+1)^2,{x,0,nn}],x ^n y],{n,2,nn}],1]  (* Geoffrey Critzer, Jan 05 2013 *)
a[ n_] := If[ n < 1, Boole[ n == 0], 2 Binomial[ 2 n, n]/(n + 1)]; (* Michael Somos, Jun 18 2014 *)
a[ n_] := SeriesCoefficient[ -1 + 4 / (1 + Sqrt[ 1 - 4 x]), {x, 0, n}]; (* Michael Somos, Jun 18 2014 *)
Table[If[n==0, 1, 2 CatalanNumber[n]], {n,0,25}] (* Peter Luschny, Feb 27 2017 *)

PARI

{a(n) = if( n<1, n==0, 2 * binomial( 2*n, n) / (n + 1))}; /* Michael Somos, Aug 17 2005 */

PARI

{a(n) = if( n<0, 0, polcoeff( -1 + 4 / (1 + sqrt(1 - 4*x + x * O(x^n))), n))}; /* Michael Somos, Aug 17 2005 */

Formula

Apart from initial term, twice Catalan numbers.

G.f.: (1 - x - sqrt(1 - 4*x)) / x. - Michael Somos, Apr 13 2012

From Paul Barry, Nov 14 2004: (Start)

G.f.: (1 + x*c(x))/(1 - x*c(x)), where c(x) is the g.f. of A000108.

a(n) = C(n)*(2-0^n), where C(n) = A000108(n).

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

Assuming offset 1, then series reversion of g.f. A(x) is -A(-x). - Michael Somos, Aug 17 2005

Assuming offset 2, then A(x) satisfies A(x - x^2) = x^2 - x^4 and so A(x) = C(x)^2 - C(x)^4, A(A(x)) = C(x)^4 - C(x)^8, A(A(A(x))) = C(x)^8 - C(x)^16, etc., where C(x) = (1-sqrt(1-4*x))/2 = x + x^2 + 2*x^3 + 5*x^4 + 14*x^5 + ... . - Paul D. Hanna, May 16 2008

Apart from initial term, INVERTi transform of A000984(n+1) = binomial(2*n+2,n+1), also, for n >= 1, a(n) = (1/Pi)*Integral_{x=0..4} x^(n-1)*sqrt(x*(4 - x)). - Groux Roland, Mar 15 2011

D-finite with recurrence (n+2)*a(n) - 2*(2*n+1)*a(n-1) = 0, n > 1. - R. J. Mathar, Nov 14 2011

For n > 0, a(n) = C(2*n+2, n+1) mod 4*C(2*n, n - 1). - Robert G. Wilson v, May 02 2012

For n > 0, a(n) = 2^(2*n + 1) * Gamma(n + 1/2)/(sqrt(Pi) * (n + 1)!). - Vaclav Kotesovec, Sep 16 2013

G.f.: 1 + 2*x/(Q(0) - x), where Q(k) = 2*x + (k + 1)/(2*k + 1) - 2*x*(k + 1)/(2*k + 1)/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, Dec 03 2013

G.f.: 3 - 4*x - 2*S(0), where S(k) = 2*k + 1 - x*(2*k + 3)/(1 - x*(2*k + 1)/S(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Dec 23 2013

0 = a(n)*(16*a(n+1) - 10*a(n+2)) + a(n+1)*(2*a(n+1) + a(n+2)) for all n in Z. - Michael Somos, Jun 18 2014

If A(x)^t = 1 + 2*t*x + Sum_{n >= 2} t*P(n,t)*x^n, then we conjecture that all the zeros of the polynomial P(n,t) lie on the vertical line Re(t) = -n/2 in the complex plane. - Peter Bala, Oct 05 2015

a(n+1) = a(n) + (1/2)*(Sum_{k=0..n} a(k)*a(n-k)) if n > 0. - Michael Somos, Apr 22 2022

b(n) = a(n+1) - a(n) for all n in Z if b(0) = 2, a(-1) = -1, a(0) = 0, a(-1) = 3, a(-2) = -1 where b = A071721. - Michael Somos, Apr 23 2022