login
The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A015084 Carlitz-Riordan q-Catalan numbers for q=3. 23
1, 1, 4, 43, 1252, 104098, 25511272, 18649337311, 40823535032644, 267924955577741566, 5274102955963545775864, 311441054994969341088610030, 55171471477692117486494217498280 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

Limit_{n->inf} a(n)/3^((n-1)(n-2)/2) = Product{k>=1} 1/(1-1/3^k) = 1.785312341998534190367486296013703535718796... - Paul D. Hanna, Jan 24 2005

It appears that the Hankel transform is 3^A002412(n). - Paul Barry, Aug 01 2008

Hankel transform of the aerated sequence is 3^C(n+1,3). - Paul Barry, Oct 31 2008

LINKS

Seiichi Manyama, Table of n, a(n) for n = 0..65

Robin Sulzgruber, The Symmetry of the q,t-Catalan Numbers, Thesis, University of Vienna, 2013.

FORMULA

a(n+1) = Sum_{i=0..n} q^i*a(i)*a(n-i) with q=3 and a(0)=1.

G.f. satisfies: A(x) = 1/(1-x*A(3*x)) = 1/(1-x/(1-3*x/(1-3^2*x/(1-3^3*x/(1-...))))) (continued fraction). - Paul D. Hanna, Jan 24 2005

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

  1,  3,  0,  0,  0,  0, ...

  1,  3,  9,  0,  0,  0, ...

  1,  3,  9, 27,  0,  0, ...

  1,  3,  9, 27, 81,  0, ...

  ... - Gary W. Adamson, Jul 14 2011

G.f.: T(0), where T(k) = 1 - x*3^k/(x*3^k - 1/T(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Oct 18 2013

EXAMPLE

G.f. = 1 + x + 4*x^2 + 43*x^3 + 1252*x^4 + 104098*x^5 + 25511272*x^6 + ...

From Seiichi Manyama, Dec 05 2016: (Start)

a(1) = 1,

a(2) = 3^1 + 1 = 4,

a(3) = 3^3 + 3^2 + 2*3^1 + 1 = 43,

a(4) = 3^6 + 3^5 + 2*3^4 + 3*3^3 + 3*3^2 + 3*3^1 + 1 = 1252. (End)

MAPLE

A015084 := proc(n)

    option remember;

    if n = 1 then

        1;

    else

    add(3^(i-1)*procname(i)*procname(n-i), i=1..n-1) ;

    end if;

end proc: # R. J. Mathar, Sep 29 2012

MATHEMATICA

a[n_] := a[n] = Sum[3^i*a[i]*a[n -i -1], {i, 0, n -1}]; a[0] = 1; Array[a, 16, 0] (* Robert G. Wilson v, Dec 24 2016 *)

m = 13; ContinuedFractionK[If[i == 1, 1, -3^(i-2) x], 1, {i, 1, m}] + O[x]^m // CoefficientList[#, x]& (* Jean-François Alcover, Nov 17 2019 *)

PROG

(PARI) a(n)=if(n==1, 1, sum(i=1, n-1, 3^(i-1)*a(i)*a(n-i))) \\ Paul D. Hanna

(Ruby)

def A(q, n)

  ary = [1]

  (1..n).each{|i| ary << (0..i - 1).inject(0){|s, j| s + q ** j * ary[j] * ary[i - 1 - j]}}

  ary

end

def A015084(n)

  A(3, n)

end # Seiichi Manyama, Dec 24 2016

CROSSREFS

Cf. A227543.

Cf. A015108 (q=-11), A015107 (q=-10), A015106 (q=-9), A015105 (q=-8), A015103 (q=-7), A015102 (q=-6), A015100 (q=-5), A015099 (q=-4), A015098 (q=-3), A015097 (q=-2), A090192 (q=-1), A000108 (q=1), A015083 (q=2), this sequence (q=3), A015085 (q=4), A015086 (q=5), A015089 (q=6), A015091 (q=7), A015092 (q=8), A015093 (q=9), A015095 (q=10), A015096 (q=11).

Column k=3 of A090182, A290759.

Sequence in context: A317140 A152282 A153255 * A176827 A220675 A222979

Adjacent sequences:  A015081 A015082 A015083 * A015085 A015086 A015087

KEYWORD

nonn

AUTHOR

Olivier Gérard

EXTENSIONS

More terms from Paul D. Hanna, Jan 24 2005

Offset changed to 0 by Seiichi Manyama, Dec 05 2016

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified July 23 14:44 EDT 2021. Contains 346259 sequences. (Running on oeis4.)