0,2

2^n*LegendreP(n,k) yields the central coefficients of (1+2kx+(k^2-1)x^2)^n, with g.f. 1/sqrt(1-4kx+4x^2) and e.g.f. exp(2kx)BesselI(0,2sqrt(k^2-1)x). - Paul Barry (pbarry(AT)wit.ie), May 25 2005

Number of Delannoy paths from (0,0) to (n,n) with steps U(0,1), H(1,0) and D(1,1) where D can have two colors. - Paul Barry (pbarry(AT)wit.ie), May 25 2005

Also number of paths from (0,0) to (n,0) using steps U=(1,1), H=(1,0) and D=(1,-1), the U steps can have three colors and H steps can have four colors. - Nour-Eddine Fahssi (fahssin(AT)yahoo.fr), Mar 31 2008

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

a(n) = 2^n*LegendreP(n, 2) = 2^n*hypergeom([ -n, n+1], [1], -1/2) = 2^n*GegenbauerC(n, 1/2, 2) = Sum_{k=0..n} 3^k*binomial(n, k)^2. Recurrence: a(n) = 4*(2*n-1)/n*a(n-1)-4*(n-1)/n*a(n-2). G.f.: 1/sqrt(1-8*x+4*x^2). - Vladeta Jovovic (vladeta(AT)Eunet.yu), May 13 2003

a(n) = central coefficient of (1+4*x+3*x^2)^n. - Paul D. Hanna (pauldhanna(AT)juno.com), Jun 03 2003

E.g.f. : exp(4x)Bessel_I(0, 2sqrt(3)x) - Paul Barry (pbarry(AT)wit.ie), Sep 20 2004

a(n)=sum{k=0..floor(n/2), C(n, k)C(2(n-k), n)(-1)^k*2^(n-2k)}; - Paul Barry (pbarry(AT)wit.ie), May 25 2005

a(n)=sum{k=0..n, C(n, k)C(n+k, k)2^(n-k)}. - Paul Barry (pbarry(AT)wit.ie), May 25 2005

a(n)=sum{k=0..n, C(n, k)^2*3^k}. - Paul Barry (pbarry(AT)wit.ie), Oct 15 2005

(PARI) a(n)=sum(k=0, n, binomial(n, k)^2*3^k)

(PARI) {a(n)=if(n<0, 0, polcoeff((1+4*x+3*x^2)^n, n))}

Cf. A001850.

Sequence in context: A007195 A067120 A143648 this_sequence A007196 A091638 A142984

Adjacent sequences: A069832 A069833 A069834 this_sequence A069836 A069837 A069838

easy,nonn

Benoit Cloitre (benoit7848c(AT)orange.fr), May 03 2002