A064613 Second binomial transform of the Catalan numbers.

1, 3, 10, 37, 150, 654, 3012, 14445, 71398, 361114, 1859628, 9716194, 51373180, 274352316, 1477635912, 8016865533, 43773564294, 240356635170, 1326359740956, 7351846397334, 40913414754324, 228508350629892

Offset: 0

Author

Karol A. Penson, Sep 24 2001

Keywords

• nonn

Comments

Exponential convolution of Catalan numbers and powers of 2. - Vladeta Jovovic, Dec 03 2004

Hankel transform of this sequence gives A000012 = [1,1,1,1,1,...]. - Philippe Deléham, Oct 24 2007

a(n) is the number of Motzkin paths of length n in which the (1,0)-steps at level 0 come in 3 colors and those at a higher level come in 4 colors. Example: a(3)=37 because, denoting U=(1,1), H=(1,0), and D=(1,-1), we have 3^3 = 27 paths of shape HHH, 3 paths of shape HUD, 3 paths of shape UDH, and 4 paths of shape UHD. - Emeric Deutsch, May 02 2011

a(n) is the number of Schroeder paths of semilength n in which the (2,0)-steps come in 2 colors and having no (2,0)-steps at levels 1,3,5,... - José Luis Ramírez Ramírez, Mar 30 2013

From Tom Copeland, Nov 08 2014: (Start)

This array is one of a family of Catalan arrays related by compositions of the special fractional linear (Möbius) transformations P(x,t)=x/(1-t*x); its inverse Pinv(x,t) = P(x,-t); and an o.g.f. of the Catalan numbers A000108, C(x) = [1-sqrt(1-4x)]/2; and its inverse Cinv(x) = x*(1-x). (Cf A126930.)

O.g.f.: G(x) = C[P[P(x,-1),-1]] = C[P(x,-2)] = (1-sqrt(1-4*x/(1-2*x)))/2 = x*A064613(x).

Ginv(x) = Pinv[Cinv(x),-2] = P[Cinv(x),2] = x(1-x)/[1+2x(1-x)] = (x-x^2)/[1+2(x-x^2)] = x - 3 x^2 + 8 x^3 - ... is -A155020(-x) ignoring first term there. (Cf. A146559, A125145.)(End)

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Isaac DeJager, Madeleine Naquin, and Frank Seidl, Colored Motzkin Paths of Higher Order, VERUM 2019.
• Francesc Fite, Kiran S. Kedlaya, Victor Rotger and Andrew V. Sutherland, Sato-Tate distributions and Galois endomorphism modules in genus 2, arXiv:1110.6638 [math.NT], 2011.
• Paveł Szabłowski, Beta distributions whose moment sequences are related to integer sequences listed in the OEIS, Contrib. Disc. Math. (2024) Vol. 19, No. 4, 85-109. See p. 98.

Crossrefs

Cf. A007317, A000108, A014318.

Cf. A125145, A146559, A126930.

Programs

Magma

I:=[3,10]; [1] cat [n le 2 select I[n] else ((8*n-2)*Self(n-1)-(12*n-12)*Self(n-2))div (n+1): n in [1..30]]; // Vincenzo Librandi, Jan 23 2017

Mathematica

CoefficientList[Series[(1-Sqrt[(1-6*x)/(1-2*x)])/2/x, {x, 0, 20}], x] (* Vaclav Kotesovec, Jun 29 2013 *)
a[n_] := 2^n Hypergeometric2F1[1/2, -n, 2, -2];
Array[a, 22, 0] (* Peter Luschny, Jan 27 2020 *)

PARI

x='x+O('x^66); Vec((1-sqrt((1-6*x)/(1-2*x)))/(2*x)) /* Joerg Arndt, Mar 31 2013 */

Formula

a(n) = Sum_{k=0..n} binomial(n, k)*binomial(2*k, k)*2^(n-k)/(k+1).

a(n) = 2^n*hypergeom([1/2, -n], [2], -2).

G.f.: (1-sqrt((1-6*x)/(1-2*x)))/(2*x). - Vladeta Jovovic, May 03 2003

With offset 1: a(1) = 1, a(n) = 2^(n-1) + Sum_{i=1..n-1} a(i)*a(n-i). - Benoit Cloitre, Mar 16 2004

D-finite with recurrence (n+1)*a(n) = (8*n-2)*a(n-1) - (12*n-12)*a(n-2). - Vladeta Jovovic, Jul 16 2004

E.g.f.: exp(4*x)*(BesselI(0, 2*x) - BesselI(1, 2*x)). - Vladeta Jovovic, Dec 03 2004

Inverse binomial transform of A104455. - Philippe Deléham, Nov 30 2007

G.f.: 1/(1-3*x-x^2/(1-4*x-x^2/(1-4*x-x^2/(1-4*x-x^2/(1-... (continued fraction). - Paul Barry, Jul 02 2009

a(n) = Sum_{0<=k<=n} A052179(n,k)*(-1)^k. - Philippe Deléham, Nov 28 2009

From Gary W. Adamson, Jul 21 2011: (Start)

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

3, 1, 0, 0, ...

1, 3, 1, 0, ...

1, 1, 3, 1, ...

1, 1, 1, 3, ...

... (End)

a(n) ~ 2^(n-3/2)*3^(n+3/2)/(n^(3/2)*sqrt(Pi)). - Vaclav Kotesovec, Jun 29 2013

G.f. A(x) satisfies: A(x) = 1/(1 - 2*x) + x * A(x)^2. - Ilya Gutkovskiy, Jun 30 2020

Extensions

Name clarified using a comment of Vladeta Jovovic by Peter Bala, Jan 27 2020