%I A005717 M1612
%S A005717 1,2,6,16,45,126,357,1016,2907,8350,24068,69576,201643,585690,1704510,
%T A005717 4969152,14508939,42422022,124191258,363985680,1067892399,3136046298,
%U A005717 9217554129,27114249960,79818194925,235128465026,693085098852
%N A005717 Construct triangle in which n-th row is obtained by expanding (1+x+x^2)^n 
               and take the next-to-central column.
%C A005717 Number of ordered trees with n+1 edges, having root of even degree and 
               nonroot nodes of outdegree at most 2. - Emeric Deutsch (deutsch(AT)duke.poly.edu), 
               Aug 02 2002
%C A005717 The connection to Motzkin numbers comes from the Lagrange inversion formula. 
               - Michael Somos, Oct 10 2003
%C A005717 Number of horizontal steps in all Motzkin paths of length n. - Emeric 
               Deutsch (deutsch(AT)duke.poly.edu), Nov 09 2003
%C A005717 Number of UHD's in all Motzkin paths of length n+2 (here U=(1,1), H=(1,
               0) and D=(1,-1)). Example: a(2)=2 because in the nine Motzkin paths 
               of length 4, HHHH, HHUD, HUDH, H(UHD), UDHH, UDUD, (UHD)H, UHHD and 
               UUDD, we have alltogether two UHD's (shown between parentheses). 
               - Emeric Deutsch (deutsch(AT)duke.poly.edu), Dec 26 2003
%C A005717 Number of ordered trees with n+1 edges, having exactly one leaf at even 
               height. Number of Dyck path of semilength n+1, having exactly one 
               peak at even height. Example: a(3)=6 because we have uuu(ud)ddd, 
               u(ud)dudud, udu(ud)dud, ududu(ud)d, u(ud)uuddd and uuudd(ud)d (here 
               u=(1,1),d=(1,-1) and the unique peak at even height is shown between 
               parentheses). - Emeric Deutsch (deutsch(AT)duke.poly.edu), Mar 10 
               2004
%C A005717 a(n)=number of Dyck (n+1)-paths containing exactly one UDU. - David Callan 
               (callan(AT)stat.wisc.edu), Jul 15 2004
%C A005717 Number of peaks in all Motzkin paths of length n+1. - Emeric Deutsch 
               (deutsch(AT)duke.poly.edu), Sep 01 2004
%C A005717 a(n) = A111808(n,n-1). - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), 
               Aug 17 2005
%C A005717 This is a kind of Motzkin transform of A059841 because the substitution 
               x -> x*A001006(x) in the independent variable of the g.f. of A059841 
               generates 1,0,1,2,6,16,... that is 1,0 followed by this sequence 
               here. [From R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Nov 08 2008]
%D A005717 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%D A005717 L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 78.
%H A005717 T. D. Noe, <a href="b005717.txt">Table of n, a(n) for n=1..200</a>
%H A005717 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               TrinomialCoefficient.html">Link to a section of The World of Mathematics.</
               a>
%F A005717 Sum{T(k, k-1)}, k = 1, 2, ..., n, where T is the array defined in A025177.
%F A005717 G.f.: 2x/[1-2x-3x^2+(1-x)sqrt(1-2x-3x^2)] - Emeric Deutsch (deutsch(AT)duke.poly.edu), 
               Aug 14 2002
%F A005717 E.g.f.: exp(x) I_1(2x), where I_1 is Bessel function. - Michael Somos, 
               Sep 09 2002.
%F A005717 a(n) = Sum[(-1)^k binomial[n,k] binomial[2n-2-3k,n-1],{k,0,Floor[(n-1)/
               3]}]. - David Callan (callan(AT)stat.wisc.edu), Jul 03 2006
%F A005717 a(n)=n*sum{k=0..floor((n-1)/2), C(n-1,2k)*C(k)}, C(n)=A000108(n); a(n)=sum{k=0..floor((n-1)/
               2), (2k+1)*C(n,2k+1)*C(k)}; a(n)=sum{k=0..n-1, sum{j=0..floor(k/2), 
               C(k,2j)*C(2j+1,j)}}; - Paul Barry (pbarry(AT)wit.ie), Feb 05 2007
%F A005717 a(n)=(A002426(n+1)-A002426(n))/2; - Paul Barry (pbarry(AT)wit.ie), May 
               22 2008
%F A005717 a(n)=n*A001006(n-1). [From Paul Barry (pbarry(AT)wit.ie), Oct 05 2009]
%p A005717 seq( sum('binomial(i,k)*binomial(i-k,k+1)', 'k'=0..floor(i/2)), i=1..30 
               ); # Detlef Pauly (dettodet(AT)yahoo.de), Nov 09 2001
%t A005717 Table[Coefficient[Expand[(1+x+x^2)^n], x, n-1], {n, 1, 40}]
%o A005717 (PARI) a(n)=if(n<0,0,polcoeff((1+x+x^2)^n,n-1))
%o A005717 (PARI) a(n)=if(n<0,0,n++; n*polcoeff(serreverse(x/(1+x+x^2)+x*O(x^n)),
               n))
%Y A005717 A diagonal of A027907. Cf. A002426.
%Y A005717 a(n)=n*A001006(n-1).
%Y A005717 Sequence in context: A055544 A126285 A026163 this_sequence A025266 A074403 
               A151391
%Y A005717 Adjacent sequences: A005714 A005715 A005716 this_sequence A005718 A005719 
               A005720
%K A005717 nonn,easy
%O A005717 1,2
%A A005717 N. J. A. Sloane (njas(AT)research.att.com).
%E A005717 More terms from Erich Friedman (efriedma(AT)stetson.edu), Jun 01 2001