0,3

More generally, coefficients of (1+m*x-sqrt(m^2*x^2-(2*m+2)*x+1))/(2*m*x) are given by : a(n)=sum(k=0,n,(m+1)^k*N(n,k)) where N(n,k)=1/n*C(n,k)*C(n,k+1) are the Narayana numbers (A001263). - Benoit Cloitre (benoit7848c(AT)orange.fr), May 24 2003

If y=x*A(x) then 3y^2-(1+2x)y+x=0 and x=y(1-3y)/(1-2y). - Michael Somos, Sep 28 2003

The sequence 0,1,4,19,... with g.f. (1-4x-sqrt(1-8x+4x^2))/(6x) and has a(n)=sum{k=0..floor((n-1)/2), C(n-1,2k)C(k)4^(n-1-2k)3^k}. a(n+1)=sum{k=0..floor(n/2), C(n,2k)C(k)4^(n-2k)3^k} counts Motzkin paths of length n in which the level steps have 4 colors and the up steps have 3. It is the binomial transform of A107264, and corresponds to the series reversion of x/(1+4x+3x^2). - Paul Barry (pbarry(AT)wit.ie), May 18 2005

The Hankel transform of this sequence is 3^C(n+1,2) . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Oct 29 2007

C. Coker, A family of eigensequences, Discrete Math. 282 (2004), 249-250.

Paul Barry, On Integer-Sequence-Based Constructions of Generalized Pascal Triangles, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.4.

T. D. Noe, Table of n, a(n) for n=0..100

M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210.

N. J. A. Sloane, Transforms

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 443

G.f.= (1+2x-sqrt(1-8x+4x^2))/(6x). - Emeric Deutsch (deutsch(AT)duke.poly.edu), Nov 03 2001

a(0)=1, for n>=1 a(n)=sum(k=0, n, 3^k*N(n, k)) where N(n, k)=1/n*C(n, k)*C(n, k+1) are the Narayana numbers (A001263). - Benoit Cloitre (benoit7848c(AT)orange.fr), May 24 2003

a(n) = Sum_{k=0..n} A088617(n, k)*3^k*(-2)^(n-k). - DELEHAM Philippe (kolotoko(AT)wanadoo.fr), Jan 21 2004

With offset 1 : a(1)=1, a(n)=-2*a(n-1)+3*sum(i=1, n-1, a(i)*a(n-i)) - Benoit Cloitre (benoit7848c(AT)orange.fr), Mar 16 2004

a(n) = [4(2n-1)a(n-1) - 4(n-2)a(n-2)] / (n+1) for n>=2, a(0) = a(1) = 1 . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Aug 19 2005

(PARI) a(n)=if(n<1, n==0, sum(k=0, n, 3^k*binomial(n, k)*binomial(n, k+1))/n) (from Michael Somos)

(PARI) a(n)=if(n<0, 0, n++; polcoeff(serreverse(x*(1-3*x)/(1-2*x)+x*O(x^n)), n)) (from Michael Somos)

Sequence in context: A006194 A047099 A083882 this_sequence A086624 A151382 A078940

Adjacent sequences: A007561 A007562 A007563 this_sequence A007565 A007566 A007567

nonn,nice,eigen

njas.