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 A015084 Carlitz-Riordan q-Catalan numbers for q=3. 30
 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 * A355081 A176827 A220675 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

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Last modified November 29 03:03 EST 2023. Contains 367422 sequences. (Running on oeis4.)