OFFSET
0,2
COMMENTS
The sequence b(n) = [0,1,2,6,21,80,322,1348,...] for n >= 0 is the Catalan transform of Catalan numbers C(n-1), with C(-1)=0; Sum_{k=0..n} A106566(n,k) * A000108(k-1) = b(n).
A121988 is an essentially identical sequence. - R. J. Mathar, Jun 13 2008
Catalan transform of A014137. - R. J. Mathar, Nov 11 2008
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..300
Elżbieta Liszewska and Wojciech Młotkowski, Some relatives of the Catalan sequence, arXiv:1907.10725 [math.CO], 2019.
FORMULA
a(n-1) = (1/n)*Sum_{k=1..n} binomial(2*n-k-1, n-1)*binomial(2*k-2, k-1).
G.f.: (1-sqrt(2*sqrt(1-4*x)-1))/(2*x). - Emeric Deutsch, Jun 20 2007 Corrected by Stefan Forcey (sforcey(AT)tnstate.edu), Aug 02 2007
From Vaclav Kotesovec, Oct 20 2012: (Start)
Recurrence: 3*n*(n+1)*a(n) = 14*n*(2*n-1)*a(n-1) - 4*(4*n-5)*(4*n-3)*a(n-2).
a(n) ~ 2^(4*n+3/2)/(3^(n+1/2)*sqrt(Pi)*n^(3/2)). (End)
0 = +a(n)*(+a(n+1)*(+262144*a(n+2) -275968*a(n+3) +52608*a(n+4)) +a(n+2)*(-50176*a(n+2) +107680*a(n+3) -27930*a(n+4)) +a(n+3)*(-6006*a(n+3) +2574*a(n+4))) +a(n+1)*(+a(n+1)*(-17920*a(n+2) +21952*a(n+3) -4494*a(n+4)) +a(n+2)*(+5152*a(n+2) -15820*a(n+3) +4611*a(n+4)) +a(n+3)*(+1470*a(n+3) -630*a(n+4))) +a(n+2)*(+a(n+2)*(+42*a(n+2) +129*a(n+3) -63*a(n+4)) +a(n+3)*(-63*a(n+3) +27*a(n+4))) for n>=0. - Michael Somos, May 28 2023
From Seiichi Manyama, Jan 10 2023: (Start)
G.f.: (1/x) * Series_Reversion( x * (1-x) * (1-x+x^2) ).
a(n) = (1/(n+1)) * Sum_{k=0..floor(n/2)} (-1)^k * binomial(n+k,k) * binomial(3*n-k+1,n-2*k). (End)
EXAMPLE
G.f. = 1 + 2*x + 6*x^2 + 21*x^3 + 80*x^4 + 322*x^5 + 1349*x^6 + ... - Michael Somos, May 28 2023
MAPLE
c := proc (x) options operator, arrow; (1/2)*(1-sqrt(1-4*x))/x end proc; G := simplify(c(x)*c(x*c(x))); Gser := series(G, x = 0, 28); seq(coeff(Gser, x, n), n = 0 .. 24) # Emeric Deutsch, Jun 20 2007
MATHEMATICA
a[n_]:= Sum[ Binomial[2n -k-1, n-1]*Binomial[2k-2, k-1], {k, n}]/n;
Array[a, 23] (* Robert G. Wilson v, Jul 18 2007 *)
PROG
(Magma) R<x>:=PowerSeriesRing(Rationals(), 40); Coefficients(R!( (1-Sqrt(2*Sqrt(1-4*x)-1))/(2*x) )); // G. C. Greubel, Feb 06 2024
(SageMath)
def A129442_list(prec):
P.<x> = PowerSeriesRing(ZZ, prec)
return P( (1-sqrt(2*sqrt(1-4*x)-1))/(2*x) ).list()
A129442_list(40) # G. C. Greubel, Feb 06 2024
CROSSREFS
KEYWORD
nonn
AUTHOR
Philippe Deléham, May 28 2007, Jun 20 2007
EXTENSIONS
More terms from Emeric Deutsch, Jun 20 2007
STATUS
approved