A007854 Expansion of 1/(1 - 3*x*C(x)), where C(x) = (1 - sqrt(1 - 4*x))/(2*x) = g.f. for the Catalan numbers A000108.

1, 3, 12, 51, 222, 978, 4338, 19323, 86310, 386250, 1730832, 7763550, 34847796, 156503064, 703149438, 3160160811, 14206181382, 63874779714, 287242041528, 1291872728826, 5810776384932, 26138647551564, 117587214581508

Offset: 0

Author

Martin Klazar

Keywords

• nonn
• easy

Comments

Chains in rooted plane trees on n nodes.

The Hankel transform of the aerated sequence with g.f. 1/(1-3x^2c(x^2)) is also 3^n. In general, the expansions of 1/(1-k*x*c(x)) and 1/(1-k*x^2*c(x^2)) have Hankel transform k^n. - Paul Barry, Jan 20 2007

Binomial transform of A112657. - Philippe Deléham, Nov 25 2007

Row sums of the Riordan matrix (1/sqrt(1-4x),(1-sqrt(1-4x))/(2*sqrt(1-4x))) (A116395). - Emanuele Munarini, Apr 26 2011

Numbers have the same parity as the Catalan numbers, that is, a(n) is even except for n of the form 2^m - 1. Follows from C(x) = 1/(1 - x*C(x)) = 1/(1 - 3*x*C(x)) (mod 2). - Peter Bala, Jul 24 2016

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• Paul Barry, On a Central Transform of Integer Sequences, arXiv:2004.04577 [math.CO], 2020.
• Martin Klazar, Twelve countings with rooted plane trees, European Journal of Combinatorics 18 (1997), 195-210; Addendum, 18 (1997), 739-740.
• J.-C. Novelli and J.-Y. Thibon, Free quasi-symmetric functions of arbitrary level, arXiv:math/0405597 [math.CO], 2004.
• Index entries for sequences related to rooted trees

Crossrefs

Cf. A000108, A000984, A076035, A076036, A067347, A116395, A126694.

Programs

Mathematica

CoefficientList[Series[(1+3Sqrt[1-4x])/(4-18x),{x,0,25}],x] (* Emanuele Munarini, Apr 26 2011 *)
nm = 25; t = NestList[Append[Accumulate[#], 3 Total[#]] &, {1}, nm];
Table[t[[n, n]], {n, nm}] (*similar to generating Catalan's triangle A009766*)
(* Li Han, Oct 23 2020 *)

Maxima

makelist(kron_delta(n,0)+sum(binomial(2*n-k,n-k)*(k*3^k)/(2*n-k),k,1,n),n,0,12); /* Emanuele Munarini, Apr 26 2011 */

Formula

a(n) = (9*a(n-1)-3*A000108(n-2))/2 = 3*A049027(n-1) = A067336(n-1)*3/2 = A049027(n-1) + A067336(n-1) = A067347(3, n-1). - Henry Bottomley, Jan 16 2002

a(n) = Sum_{k>=0} A106566(n, k)*3^k. - Philippe Deléham, Aug 11 2005

The Hankel transform of this sequence is A000244 = [1, 3, 9, 27, 81, 243, 729, ...](powers of 3). - Philippe Deléham, Nov 26 2006

a(n) = Sum_{k = 0..n} C(2n,n-k)(2k+1)2^k/(n+k+1). - Paul Barry, Jan 20 2007

a(n) = Sum_{k = 0..n} A039599(n,k)*2^k. - Philippe Deléham, Sep 08 2007

a(n) = Sum_{k = 0..n} A116395(n,k). - Vladimir Kruchinin, Mar 09 2011

From Emanuele Munarini, Apr 26 2011 (Start)

a(n) = Sum_{k = 1..n} C(2*n-k,n-k)*(k*3^k)/(2*n-k), for n>0.

a(n) = (1/4)*(9/2)^n-3*Sum_{k=0..n} C(2*k,k)/(2k-1)*(9/2)^(n-k).

D-finite with recurrence: 2*(n+2)*a(n+2)-(17*n+22)*a(n+1)+18*(2*n+1)*a(n)=0. (End)

From Gary W. Adamson, Jul 14 2011: (Start)

a(n) = upper left term in M^n, M = the infinite square production matrix:

3, 3, 0, 0, 0, 0, ...

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

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

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

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

... (End)

Extensions

More terms from Henry Bottomley, Jan 16 2002