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A084603 Coefficients of 1/sqrt(1 - 2*x - 11*x^2); also, a(n) is the central coefficient of (1 + x + 3*x^2)^n.

%I A084603 #56 Apr 23 2025 04:50:07
%S A084603 1,1,7,19,91,331,1441,5797,24739,103411,441397,1876777,8047909,
%T A084603 34533253,148803487,642228139,2778852979,12043194163,52286516821,
%U A084603 227323871929,989675651041,4313712072241,18822940658947,82215245701519
%N A084603 Coefficients of 1/sqrt(1 - 2*x - 11*x^2); also, a(n) is the central coefficient of (1 + x + 3*x^2)^n.
%C A084603 5th binomial transform of 2^n*LegendreP(n,-2) (signed version of A069835). - _Paul Barry_, 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. - _N-E. Fahssi_, Feb 05 2008
%C A084603 Diagonal of the rational function 1 / (1 - 3*x^2 - y^2 - x*y). - _Ilya Gutkovskiy_, Apr 22 2025
%H A084603 Seiichi Manyama, <a href="/A084603/b084603.txt">Table of n, a(n) for n = 0..1542</a> (terms 0..200 from Vincenzo Librandi)
%H A084603 Tony D. Noe, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL9/Noe/noe35.html">On the Divisibility of Generalized Central Trinomial Coefficients</a>, 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_, Aug 26 2004
%F A084603 Binomial transform is A084609. Hankel transform is 6^n*3^C(n,2). - _Paul Barry_, Sep 16 2006
%F A084603 a(n) = (1/Pi)*Integral_{x=1-2*sqrt(3)..1+2*sqrt(3)} x^n/sqrt(-x^2 + 2*x + 11). - _Paul Barry_, Sep 16 2006
%F A084603 From _Paul Barry_, Sep 16 2006: (Start)
%F A084603 a(n) = Sum_{k=0..floor(n/2)} C(n,2k)*C(2k,k)*3^k;
%F A084603 a(n) = Sum_{k=0..floor(n/2)} C(n,k)*C(n-k,k)*3^k. (End)
%F A084603 From _N-E. Fahssi_, Feb 05 2008: (Start)
%F A084603 a(n) is also the central coefficient of (3+x+x^2)^n;
%F A084603 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. (End)
%F A084603 D-finite with recurrence: a(n+2) = ( (2*n+3)*a(n+1) + 11*(n+1)*a(n) )/(n+2); a(0)=a(1)=1. - _Sergei N. Gladkovskii_, Aug 01 2012
%F A084603 a(n) ~ sqrt(18+3*sqrt(3))*(1+2*sqrt(3))^n/(6*sqrt(Pi*n)). - _Vaclav Kotesovec_, Oct 14 2012
%F A084603 E.g.f.: exp(x)*BesselI(0, 2*sqrt(3)*x). - _Paul D. Hanna_, Nov 09 2014, after _Vladeta Jovovic_ in A084601
%F A084603 From _Peter Bala_, Jan 07 2022: (Start)
%F A084603 O.g.f. A(x) = 1 + x*d/dx(log(B(x))), where B(x) = (1 - x - sqrt(1 - 2*x - 11*x^2))/(6*x^2) is the o.g.f. of A025237.
%F A084603 The Gauss congruences a(n*p^k) == a(n^p^(k-1)) (mod p^k) hold for prime p and positive integers n and k. (End)
%t A084603 Table[Sum[Binomial[n-k,k]*Binomial[n,k]*3^k,{k,0,Floor[n/2]}],{n,0,20}] (* _Vaclav Kotesovec_, Oct 14 2012 *)
%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, A025237.
%K A084603 nonn,easy
%O A084603 0,3
%A A084603 _Paul D. Hanna_, Jun 01 2003