0,2

a(n) = number of Delannoy paths (A001850) from (0,0) to (n,n) in which every Northeast step is immediately preceded by an East step. - David Callan (callan(AT)stat.wisc.edu), Mar 14 2004

The Hankel transform (see A001906 for definition) of this sequence is A036442 : 1, 4, 32, 512, 16384, ... . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Jul 03 2005

In general, 1/sqrt(1-4*r*x-4*r*x^2) has e.g.f. exp(2rx)BesselI(0,2r*sqrt((r+1)/r)x), a(n)=sum{k=0..n, C(2k,k)C(k,n-k)r^k}, gives the central coefficient of (1+(2r)x+r(r+1)x^2) and is the (2r)-th binomial transform of 1/sqrt(1-8*C(n+1,2)x^2). - Paul Barry (pbarry(AT)wit.ie), Apr 28 2005

Also number of paths from (0,0) to (n,0) using steps U=(1,1), H=(1,0) and D=(1,-1), the H and U steps can have two colors. - Nour-Eddine Fahssi (fahssin(AT)yahoo.fr), Feb 05 2008

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

D. Castellanos, A generalization of Binet's formula and some of its consequences, Fib. Quart., 27 (1989), 424-438.

Tony D. Noe, On the Divisibility of Generalized Central Trinomial Coefficients, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.7.

T. D. Noe, Table of n, a(n) for n = 0..200

Sum(binomial(2*k, k)*binomial(k, n-k), k=0..n). - Detlef Pauly (dettodet(AT)yahoo.de), Nov 08 2001

G.f.: 1/(1-4x-4x^2)^(1/2); also, a(n) is the central coefficient of (1+2x+2x^2)^n. - Paul D. Hanna (pauldhanna(AT)juno.com), Jun 01 2003

Inverse binomial transform of central Delannoy numbers A001850. - David Callan (callan(AT)stat.wisc.edu), Mar 14 2004

E.g.f.: exp(2*x)*BesselI(0, 2*sqrt(2)*x). - Vladeta Jovovic (vladeta(AT)eunet.rs), Mar 21 2004

a(n)=sum{k=0..floor(n/2), C(n,2k)C(2k,k)2^(n-k)}. - Paul Barry (pbarry(AT)wit.ie), Sep 19 2006

seq( sum('binomial(2*k, k)*binomial(k, n-k)', 'k'=0..n), n=0..30 ); # Detlef Pauly (dettodet(AT)yahoo.de), Nov 08 2001

(PARI) for(n=0, 30, t=polcoeff((1+2*x+2*x^2)^n, n, x); print1(t", "))

Cf. A002426, A084600-A084606, A084608-A084615.

Cf. A106258, A106259, A106260, A106261.

First column of A110446.

Sequence in context: A150830 A150831 A084607 this_sequence A150832 A150833 A150834

Adjacent sequences: A006136 A006137 A006138 this_sequence A006140 A006141 A006142

nonn,easy

N. J. A. Sloane (njas(AT)research.att.com).