Search: id:A005773 Results 1-1 of 1 results found. %I A005773 M1443 %S A005773 1,1,2,5,13,35,96,267,750,2123,6046,17303,49721,143365,414584,1201917, %T A005773 3492117,10165779,29643870,86574831,253188111,741365049,2173243128, %U A005773 6377181825,18730782252,55062586341,161995031226,476941691177 %N A005773 Number of directed animals of size n (or directed n-ominoes in standard position). %C A005773 Sequence, with first term a(0) deleted, appears to be determined by conditions that diagonal and first superdiagonal of U are {1,1,1,1,...} and {2,3,4,5,...,n+1,...}, where A=LU is LU factorization of Hankel matrix A given by [{a(1),a(2),...},{a(2),a(3),...},...,{a(n),a(n+1),...}, ...]- John W. Layman (layman(AT)math.vt.edu), Jul 21 2000 %C A005773 Also the number of base 3 n-digit numbers with digit sum n. For the analogous sequence in base 10 see A071976. - John W. Layman (layman(AT)math.vt.edu), Jun 22 2002 %C A005773 Also number of paths in an n X n grid from (0,0) to the line x=n-1, using only steps U=(1,1), H=(1,0) and D=(1,-1) (i.e. left factors of length n-1 of Motzkin paths). Example: a(3)=5, namely, HH, UD, HU, UH and UU. Also number of ordered trees with n edges and having nonroot nodes of outdegree at most 2. - Emeric Deutsch (deutsch(AT)duke.poly.edu), Aug 01 2002 %C A005773 Number of symmetric Dyck paths of semilength 2n-1 with no peaks at even level. Example: a(3)=5 because we have UDUDUDUDUD, UDUUUDDDUD, UUUUUDDDDD, UUUDUDUDDD and UUUDDUUDDD, where U=(1,1) and D=(1,-1). Also number of symmetric Dyck paths of semilength 2n with no peaks at even level. Example: a(3)=5 because we have UDUDUDUDUDUD, UDUUUDUDDDUD, UUUDUDUDUDDD, UUUUUDUDDDDD and UUUDDDUUUDDD. - Emeric Deutsch (deutsch(AT)duke.poly.edu), Nov 21 2003 %C A005773 a(n)=sum of (n-1)-st central trinomial coefficient and its predecessor. Example: a(4)=6+7 and (1+x+x^2)^3=...+ 6*x^2 + 7*x^3 +... . - David Callan (callan(AT)stat.wisc.edu), Feb 07 2004 %C A005773 a(n)=number of UDU-free paths of n upsteps (U) and n downsteps (D) that start U (n>=1). Example. a(2)=2 counts UUDD, UDDU. - David Callan (callan(AT)stat.wisc.edu), Aug 18 2004 %C A005773 Hankel transform of a(n+1) = [1,2,5,13,35,96,...]gives A000012 = [1,1, 1,1,1,1,...] . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Oct 24 2007 %C A005773 Equals row sums of triangle A136787 starting (1, 2, 5, 13, 35,...). - Gary W. Adamson (qntmpkt(AT)yahoo.com), Jan 21 2008 %C A005773 a(n) = number of permutations on [n] that avoid the patterns 1-23-4 and 1-3-2, where the omission of a dash in a pattern means the permutation entries must be adjacent. Example: a(4) = 13 counts all 14 (Catalan number) (1-3-2)-avoiding permutations on [4] except 1234. - David Callan (callan(AT)stat.wisc.edu), Jul 22 2008 %C A005773 Contribution from Eric Egge (eegge(AT)carleton.edu), Oct 21 2008: (Start) %C A005773 a(n) is also the number of involutions of length 2n-2 which are %C A005773 invariant under the reverse-complement map and have no decreasing %C A005773 subsequences of length 4. (End) %C A005773 Hankel transform is A010892. [From Paul Barry (pbarry(AT)wit.ie), Jan 19 2009] %C A005773 Starting (1, 2, 5, 13,...) = row sums of triangle A158793 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Mar 26 2009] %C A005773 a(n) = the number of Dyck words of semilength n with no DUUU. For example, a(4) = 14-1 = 13 because there is only one Dyck 4-word containing DUUU, namely UDUUUDDD. [From Eric S Rowland (erowland(AT)math.rutgers.edu), Apr 21 2009] %D A005773 E. Barcucci et al., From Motzkin to Catalan Permutations, Discr. Math., 217 (2000), 33-49. %D A005773 M. Bousquet-M\'{e}lou, New enumerative results on two-dimensional directed animals, Discr. Math., 180 (1998), 73-106. %D A005773 D. Dhar et al., Enumeration of directed site animals on two-dimensional lattices, J. Phys. A 15 (1982), L279-L284. %D A005773 J. E. Goodman and J. O'Rourke, editors, Handbook of Discrete and Computational Geometry, CRC Press, 1997, p. 237. %D A005773 D. Gouyou-Beauchamps and G. Viennot, Equivalence of the two-dimensional directed animal problem to a one-dimensional path problem, Adv. in Appl. Math. 9 (1988), no. 3, 334-357. %D A005773 T. Mansour, Restricted 1-3-2 permutations and generalized patterns, Annals of Combin., 6 (2002), 65-76. %D A005773 J. Nemecek and M. Klazar, A bijection between nonnegative words and sparse abba-free partitions, Discr. Math., 265 (2003), 411-416. %D A005773 A. Sapounakis, I. Tasoulas and P. Tsikouras, Counting strings in Dyck paths, Discrete Math., 307 (2007), 2909-2924. %D A005773 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence). %D A005773 R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Problem 6.46a. %H A005773 T. D. Noe, Table of n, a(n) for n=0..200 %H A005773 H. Bottomley, Illustration of initial terms %H A005773 M. Bousquet-M\'{e}lou, New enumerative results on two-dimensional directed animals %H A005773 E. Deutsch and B. E. Sagan, Congruences for Catalan and Motzkin numbers and related sequences, J. Num. Theory 117 (2006), 191-215. %H A005773 P. Flajolet and R. Sedgewick, Analytic Combinatorics, 2009; see page 81 %H A005773 J. W. Layman, The Hankel Transform and Some of its Properties, J. Integer Sequences, 4 (2001), #01.1.5. %H A005773 T. Mansour, Restricted 1-3-2 permutations and generalized patterns. %H A005773 P. Peart and W.-J. Woan, Generating Functions via Hankel and Stieltjes Matrices, J. Integer Seqs., Vol. 3 (2000), #00.2.1. %F A005773 G.f.: 2x/(3x-1+sqrt(1-2x-3x^2)) - Len Smiley (smiley(AT)math.uaa.alaska.edu). %F A005773 Also a(0)=1, a(n) = M(n) + Sum {M(k)*a(n-k-1), k=0..n-1}, where M(n) are the Motzkin numbers (A001006). %F A005773 na(n)=2na(n-1)+3(n-2)a(n-2), a(0)=a(1)=1. - Michael Somos, Feb 02, 2002. %F A005773 G.f.: (1/2)((1+x)/(1-3x))^(1/2) + 1/2 . Related to Motzkin numbers A001006 by a(n+1, 0) = 3 a(n) - A001006(n). %F A005773 a(n)=sum(binomial(q, floor(q/2))binomial(n-1, q), q=0..n) for n>0 - Emeric Deutsch (deutsch(AT)duke.poly.edu), Aug 15 2002 %F A005773 a(n+1)=sum{k=0..n, (-1)^(n+k)C(n, k)C(2k+1, k+1)}; a(n)=0^n+sum{k=0..n-1, (-1)^(n+k-1)C(n-1, k)C(2k+1, k+1)}. - Paul Barry (pbarry(AT)wit.ie), Jun 22 2004 %F A005773 a(n+1)=sum{k=0..n, (-1)^k*3^(n-k)*binomial(n, k)A000108(k)} - Paul Barry (pbarry(AT)wit.ie), Jan 27 2005 %F A005773 Starting (1, 2, 5, 13,...) gives binomial transform of A001405 and inverse binomial transform of A001700. - Gary W. Adamson (qntmpkt(AT)yahoo.com), Aug 31 2007 %F A005773 Starting (1, 2, 5, 13, 35, 96,...) gives row sums of triangle A132814. - Gary W. Adamson (qntmpkt(AT)yahoo.com), Aug 31 2007 %F A005773 Contribution from Paul Barry (pbarry(AT)wit.ie), Jan 19 2009: (Start) %F A005773 G.f.: 1/(1-x/(1-x-x^2/(1-x-x^2/(1-x-x^2/(1-x-x^2/(1-x-x^2/(1-... (continued fraction). %F A005773 G.f.: 1+x/(1-2x-x^2/(1-x-x^2/(1-x-x^2/(1-x-x^2/(1-.... (continued fraction). (End) %F A005773 Formula from Thomas Wieder (wieder.thomas(AT)t-online.de), Feb 25 2009: %F A005773 a(n) = sum_{l_1=0}^{n+1} sum_{l_2=0}^{n}...sum_{l_i=0}^{n-i}...sum_{l_n=0}^{1} %F A005773 delta(l_1,l_2,...,l_i,...,l_n) %F A005773 where delta(l_1,l_2,...,l_i,...,l_n) = 0 if any (l_i - l_(i+1))^2 >= 2 for i=1.. n-1 %F A005773 and delta(l_1,l_2,...,l_i,...,l_n) = 1 otherwise. %F A005773 INVERT transform of offset Motzkin numbers (A001006): (a(n))_{n>=1}=(1, 1,2,4,9,21,...). [From David Callan (callan(AT)stat.wisc.edu), Aug 27 2009] %p A005773 seq( sum('binomial(i-1,k)*binomial(i-k,k)', 'k'=0..floor(i/2)), i=0..30 ); # Detlef Pauly (dettodet(AT)yahoo.de), Nov 09 2001 %p A005773 Contribution from Thomas Wieder (thomas.wieder(AT)t-online.de), Feb 22 2009: (Start) %p A005773 A005773:=proc(n::integer) %p A005773 local i,j,A,istart,iend,KartProd,Liste,Term,delta; %p A005773 A:=0; %p A005773 for i from 0 to n do %p A005773 Liste[i]:=NULL; %p A005773 istart[i]:=0; %p A005773 iend[i]:=n-i+1: %p A005773 for j from istart[i] to iend[i] do %p A005773 Liste[i]:=Liste[i],j; %p A005773 end do; %p A005773 Liste[i]:=[Liste[i]]: %p A005773 end do; %p A005773 KartProd:=cartprod([seq(Liste[i],i=1..n)]); %p A005773 while not KartProd[finished] do %p A005773 Term:=KartProd[nextvalue](); %p A005773 delta:=1; %p A005773 for i from 1 to n-1 do %p A005773 if (op(i,Term) - op(i+1,Term))^2 >= 2 then %p A005773 delta:=0; %p A005773 break; %p A005773 end if; %p A005773 end do; %p A005773 A:=A+delta; %p A005773 end do; %p A005773 end proc; %p A005773 (End) %o A005773 (PARI) a(n)=if(n<2,n>=0,(2*n*a(n-1)+3*(n-2)*a(n-2))/n) %Y A005773 See also A005775. Inverse of A001006. Also sum of numbers in row n+1 of array T in A026300. Leading column of array in A038622. %Y A005773 The right edge of the triangle A062105. %Y A005773 Interpolates between Motzkin numbers (A001006) and Catalan numbers (A000108). Cf. A054391, A054392, A054393, A055898. %Y A005773 Except for the first term a(0), sequence is the binomial transform of A001405. %Y A005773 a(n) = A002426(n-1)+A005717(n-1) if n>0. - Emeric Deutsch (deutsch(AT)duke.poly.edu), Aug 14 2002 %Y A005773 Cf. A001405, A001700, A132814. %Y A005773 Cf. A136787. %Y A005773 A158973 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Mar 26 2009] %Y A005773 Sequence in context: A000107 A063028 A085810 this_sequence A022855 A091190 A007689 %Y A005773 Adjacent sequences: A005770 A005771 A005772 this_sequence A005774 A005775 A005776 %K A005773 nonn,easy,nice %O A005773 0,3 %A A005773 N. J. A. Sloane (njas(AT)research.att.com), Simon Plouffe, Clark Kimberling (ck6(AT)evansville.edu) %E A005773 More terms from David W. Wilson (davidwwilson(AT)comcast.net) Search completed in 0.002 seconds