A104455 Expansion of e.g.f. exp(5*x)*(BesselI(0,2*x) - BesselI(1,2*x)).

1, 4, 17, 77, 371, 1890, 10095, 56040, 320795, 1881524, 11250827, 68330773, 420314629, 2612922694, 16389162537, 103587298965, 659071002195, 4217699773140, 27129590096595, 175303621195647, 1137400502295081, 7406899253418414, 48396105031873197, 317180187174490902, 2084542632685363221

Offset: 0

Author

Paul Barry, Mar 08 2005

Keywords

• easy
• nonn

Comments

Third binomial transform of A000108. In general, the k-th binomial transform of A000108 will have g.f. (1-sqrt((1-(k+4)*x)/(1-k*x)))/(2*x), e.g.f. exp((k+2)*x)*(BesselI(0,2*x) - BesselI(1,2*x)) and a(n) = Sum_{i=0..n} C(n,i)*C(i)*k^(n-i).

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

In general, the k-th binomial transform of A000108 can be generated from M^n, M = the production matrix of the form shown in the formula section, with a diagonal (k+1, k+1, k+1, ...). - Gary W. Adamson, Jul 21 2011

a(n) is the number of Schroeder paths of semilength n in which the H=(2,0) steps come in 3 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(1-4*x)]/2; and its inverse Cinv(x) = x*(1-x). (Cf A091867.)

O.g.f.: G(x) = C[P[P[P(x,-1),-1]]-1] = C[P(x,-3)] = [1-sqrt(1-4*x/(1-3*x)]/2 = x*A104455(x).

Ginv(x) = Pinv[Cinv(x),-3]= P[Cinv(x),3] = x*(1-x)/[1+3*x*(1-x)] = (x-x^2)/[1+3(x-x^2)] = x*A125145(-x). (Cf. A030528.) (End)

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..200

Crossrefs

Cf. A007317, A064613.

Cf. A000108, A091867, A125145, A030528.

Programs

Mathematica

CoefficientList[Series[(1-Sqrt[(1-7*x)/(1-3*x)])/(2*x), {x, 0, 20}], x] (* Vaclav Kotesovec, Oct 17 2012 *)

PARI

x='x+O('x^66); Vec((1-sqrt((1-7*x)/(1-3*x)))/(2*x)) \\ Joerg Arndt, Mar 31 2013

Formula

G.f.: (1-sqrt((1-7*x)/(1-3*x)))/(2*x).

a(n) = Sum_{k=0..n} C(n, k)*C(k)*3^(n-k).

a(n) = 3^n+Sum_{k=0..n-1} a(k)*a(n-1-k), a(0)=1. - Philippe Deléham, Dec 12 2009

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

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

4, 1, 0, 0, ...

1, 4, 1, 0, ...

1, 1, 4, 1, ...

1, 1, 1, 4, ...

(End)

D-finite with recurrence: (n+1)*a(n) = 2*(5*n-1)*a(n-1) - 21*(n-1)*a(n-2). - Vaclav Kotesovec, Oct 17 2012

a(n) ~ 7^(n+3/2)/(8*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Oct 17 2012

G.f. A(x) satisfies: A(x) = 1/(1 - 3*x) + x * A(x)^2. - Ilya Gutkovskiy, Jun 30 2020