A129442 Expansion of c(x)*c(x*c(x)) where c(x) is the g.f. of A000108.

1, 2, 6, 21, 80, 322, 1348, 5814, 25674, 115566, 528528, 2449746, 11485068, 54377288, 259663576, 1249249981, 6049846848, 29469261934, 144293491564, 709806846980, 3506278661820, 17385618278700, 86500622296800

Offset: 0

Author

Philippe Deléham, May 28 2007, Jun 20 2007

Keywords

• nonn

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

Examples

			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
		

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.

Crossrefs

Cf. A000108, A014137, A106566, A121988, A236339.

Programs

Magma

R:=PowerSeriesRing(Rationals(), 40); Coefficients(R!( (1-Sqrt(2*Sqrt(1-4*x)-1))/(2*x) )); // G. C. Greubel, Feb 06 2024

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 *)

SageMath

def A129442_list(prec):
    P. = 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

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)

Extensions

More terms from Emeric Deutsch, Jun 20 2007