%I A084603
%S A084603 1,1,7,19,91,331,1441,5797,24739,103411,441397,1876777,8047909,34533253,
%T A084603 148803487,642228139,2778852979,12043194163,52286516821,227323871929,
%U A084603 989675651041,4313712072241,18822940658947,82215245701519
%N A084603 Coefficients of 1/(1-2x-11x^2)^(1/2); also, a(n) is the central coefficient 
               of (1+x+3x^2)^n.
%C A084603 5th binomial transform of 2^n*LegendreP(n,-2) (signed version of A069835). 
               - Paul Barry (pbarry(AT)wit.ie), Sep 03 2004
%C A084603 Also number of paths from (0,0) to (n,0) using steps U=(1,1), H=(1,0) 
               and D=(1,-1), the U (or D) steps come in three colors. - Nour-Eddine 
               Fahssi (fahssin(AT)yahoo.fr), Feb 05 2008
%D A084603 Tony D. Noe, On the Divisibility of Generalized Central Trinomial Coefficients, 
               Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.7.
%F A084603 a(n)=sum{k=0..floor(n/2), binomial(n-k, k)binomial(n, k)3^k}. - Paul 
               Barry (pbarry(AT)wit.ie), Aug 26 2004
%F A084603 Binomial transform is A084609. Hankel transform is 6^n*3^C(n,2). - Paul 
               Barry (pbarry(AT)wit.ie), Sep 16 2006
%F A084603 a(n)=(1/pi)*Int(x^n/sqrt(-x^2+2x+11),x,1-2sqrt(3),1+2sqrt(3)); - Paul 
               Barry (pbarry(AT)wit.ie), Sep 16 2006
%F A084603 a(n)=sum{k=0..floor(n/2), C(n,2k)*C(2k,k)*3^k}; a(n)=sum{k=0..floor(n/
               2), C(n,k)*C(n-k,k)*3^k}. - Paul Barry (pbarry(AT)wit.ie), Sep 16 
               2006
%F A084603 a(n) is also the central coefficient of (3+x+x^2)^n; a(n)=sum_{k=0..n} 
               2^(n-k) C(n,k) T(k,n), where T(k,n) is the triangle of trinomial 
               coefficients = Coefficient of x^n of (1+x+x^2)^k : A027907 - Nour-Eddine 
               Fahssi (fahssin(AT)yahoo.fr), Feb 05 2008
%o A084603 (PARI) for(n=0,30,t=polcoeff((1+x+3*x^2)^n,n,x); print1(t","))
%Y A084603 Cf. A002426, A084600-A084602, A084604-A084615.
%Y A084603 Sequence in context: A088988 A109879 A109880 this_sequence A088883 A026574 
               A091149
%Y A084603 Adjacent sequences: A084600 A084601 A084602 this_sequence A084604 A084605 
               A084606
%K A084603 nonn
%O A084603 0,3
%A A084603 Paul D. Hanna (pauldhanna(AT)juno.com), Jun 01 2003