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A005773 Number of directed animals of size n (or directed n-ominoes in standard position).

Original entry on oeis.org

1, 1, 2, 5, 13, 35, 96, 267, 750, 2123, 6046, 17303, 49721, 143365, 414584, 1201917, 3492117, 10165779, 29643870, 86574831, 253188111, 741365049, 2173243128, 6377181825, 18730782252, 55062586341, 161995031226, 476941691177, 1405155255055, 4142457992363
Offset: 0

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Author

N. J. A. Sloane, Simon Plouffe, Clark Kimberling

Keywords

Comments

This sequence, with first term a(0) deleted, appears to be determined by the conditions that the diagonal and first superdiagonal of U are {1,1,1,1,...} and {2,3,4,5,...,n+1,...} respectively, where A=LU is the LU factorization of the Hankel matrix A given by [{a(1),a(2),...}, {a(2),a(3),...}, ..., {a(n),a(n+1),...}, ...]. - John W. Layman, Jul 21 2000
Also the number of base 3 n-digit numbers (not starting with 0) with digit sum n. For the analogous sequence in base 10 see A071976, see example. - John W. Layman, Jun 22 2002
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, palindromic Motzkin paths of length 2n-2 or 2n-1). 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, Aug 01 2002
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, Nov 21 2003
a(n) is the sum of the (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, Feb 07 2004
a(n) is the 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, Aug 18 2004
a(n) is also the number of Grand-Dyck paths of semilength n starting with an up-step and avoiding the pattern DUD. - David Bevan, Nov 19 2019
Hankel transform of a(n+1) = [1,2,5,13,35,96,...] gives A000012 = [1,1,1,1,1,1,...]. - Philippe Deléham, Oct 24 2007
Equals row sums of triangle A136787 starting (1, 2, 5, 13, 35, ...). - Gary W. Adamson, Jan 21 2008
a(n) is the 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, Jul 22 2008
a(n) is also the number of involutions of length 2n-2 which are invariant under the reverse-complement map and have no decreasing subsequences of length 4. - Eric S. Egge, Oct 21 2008
Hankel transform is A010892. - Paul Barry, Jan 19 2009
a(n) is 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. - Eric Rowland, Apr 21 2009
Inverse binomial transform of A024718. - Philippe Deléham, Dec 13 2009
Let w(i, j, n) denote walks in N^2 which satisfy the multivariate recurrence
w(i, j, n) = w(i - 1, j, n - 1) + w(i, j - 1, n - 1) + w(i + 1, j - 1,n - 1) with boundary conditions w(0,0,0) = 1 and w(i,j,n) = 0 if i or j or n is < 0. Let alpha(n) the number of such walks of length n, alpha(n) = Sum_{i = 0..n, j=0..n} w(i, j, n). Then a(n+1) = alpha(n). - Peter Luschny, May 21 2011
Number of length-n strings [d(0),d(1),d(2),...,d(n-1)] where 0 <= d(k) <= k and abs(d(k) - d(k-1)) <= 1 (smooth factorial numbers, see example). - Joerg Arndt, Nov 10 2012
a(n) is the number of n-multisets of {1,...,n} containing no pair of consecutive integers (e.g., 111, 113, 133, 222, 333 for n=3). - David Bevan, Jun 10 2013
a(n) is also the number of n-multisets of [n] in which no integer except n occurs exactly once (e.g., 111, 113, 222, 223, 333 for n=3). - David Bevan, Nov 19 2019
Number of minimax elements in the affine Weyl group of the Lie algebra so(2n+1) or the Lie algebra sp(2n). See Panyushev 2005. Cf. A245455. - Peter Bala, Jul 22 2014
The shifted, signed array belongs to an interpolated family of arrays associated to the Catalan A000108 (t=1), and Riordan, or Motzkin sums A005043 (t=0), with the interpolating (here t=-2) o.g.f. G(x,t) = (1-sqrt(1-4x/(1+(1-t)x)))/2 and inverse o.g.f. Ginv(x,t) = x(1-x)/(1+(t-1)x(1-x)) (A057682). See A091867 for more info on this family. - Tom Copeland, Nov 09 2014
Alternatively, this sequence corresponds to the number of positive walks with n steps {-1,0,1} starting at the origin, ending at any altitude, and staying strictly above the x-axis. - David Nguyen, Dec 01 2016
Let N be a squarefree number with n prime factors: p_1 < p_2 < ... < p_n. Let D be its set of divisors, E the subset of D X D made of the (d_1, d_2) for which, provided that we know which p_i are in d_1, which p_i are in d_2, d_1 <= d_2 is provable without needing to know the numerical values of the p_i. It appears that a(n+1) is the number of (d_1, d_2) in E such that d_1 and d_2 are coprime. - Luc Rousseau, Aug 21 2017
Number of ordered rooted trees with n non-root nodes and all non-root nodes having outdegrees 1 or 2. - Andrew Howroyd, Dec 04 2017
a(n) is the number of compositions (ordered partitions) of n where there are A001006(k-1) sorts of part k (see formula by Andrew Howroyd, Dec 04 2017). - Joerg Arndt, Jan 26 2024

Examples

			G.f. = 1 + x + 2*x^2 + 5*x^3 + 13*x^4 + 35*x^5 + 96*x^6 + 267*x^7 + ...
a(3) = 5, a(4) = 13; since the top row of M^3 = (5, 5, 2, 1, ...)
From _Eric Rowland_, Sep 25 2021: (Start)
There are a(4) = 13 directed animals of size 4:
  O
  O    O    O    OO              O         O
  O    O    OO   O    OO   O    OO   OOO   O    O    OO    O
  O    OO   O    O    OO   OOO  O    O    OO   OOO  OO   OOO  OOOO
(End)
From _Joerg Arndt_, Nov 10 2012: (Start)
There are a(4)=13 smooth factorial numbers of length 4 (dots for zeros):
[ 1]   [ . . . . ]
[ 2]   [ . . . 1 ]
[ 3]   [ . . 1 . ]
[ 4]   [ . . 1 1 ]
[ 5]   [ . . 1 2 ]
[ 6]   [ . 1 . . ]
[ 7]   [ . 1 . 1 ]
[ 8]   [ . 1 1 . ]
[ 9]   [ . 1 1 1 ]
[10]   [ . 1 1 2 ]
[11]   [ . 1 2 1 ]
[12]   [ . 1 2 2 ]
[13]   [ . 1 2 3 ]
(End)
From _Joerg Arndt_, Nov 22 2012: (Start)
There are a(4)=13 base 3 4-digit numbers (not starting with 0) with digit sum 4:
[ 1]   [ 2 2 . . ]
[ 2]   [ 2 1 1 . ]
[ 3]   [ 1 2 1 . ]
[ 4]   [ 2 . 2 . ]
[ 5]   [ 1 1 2 . ]
[ 6]   [ 2 1 . 1 ]
[ 7]   [ 1 2 . 1 ]
[ 8]   [ 2 . 1 1 ]
[ 9]   [ 1 1 1 1 ]
[10]   [ 1 . 2 1 ]
[11]   [ 2 . . 2 ]
[12]   [ 1 1 . 2 ]
[13]   [ 1 . 1 2 ]
(End)

References

  • J. E. Goodman and J. O'Rourke, editors, Handbook of Discrete and Computational Geometry, CRC Press, 1997, p. 237.
  • T. Mansour, Combinatorics of Set Partitions, Discrete Mathematics and Its Applications, CRC Press, 2013, p. 377.
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
  • R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Problem 6.46a.
  • R. P. Stanley, Catalan Numbers, Cambridge, 2015, p. 132.

Links

Crossrefs

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.
The right edge of the triangle A062105.
Column k=3 of A295679.
Interpolates between Motzkin numbers (A001006) and Catalan numbers (A000108). Cf. A054391, A054392, A054393, A055898.
Except for the first term a(0), sequence is the binomial transform of A001405.
a(n) = A002426(n-1) + A005717(n-1) if n > 0. - Emeric Deutsch, Aug 14 2002
Cf. A001405, A001700, A132814, A245455.
Cf. A005043, A005717, A057682, A064189, A091867.

Programs

  • Haskell
    a005773 n = a005773_list !! n
a005773_list = 1 : f a001006_list [] where
   f (x:xs) ys = y : f xs (y : ys) where
     y = x + sum (zipWith (*) a001006_list ys)
-- Reinhard Zumkeller, Mar 30 2012
  • Magma
    R:=PowerSeriesRing(Rationals(), 30); Coefficients(R!( 2*x/(3*x-1+Sqrt(1-2*x-3*x^2)) )); // G. C. Greubel, Apr 05 2019
  • Maple
    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
A005773:=proc(n::integer)
local i, j, A, istart, iend, KartProd, Liste, Term, delta;
    A:=0;
    for i from 0 to n do
        Liste[i]:=NULL;
        istart[i]:=0;
        iend[i]:=n-i+1:
        for j from istart[i] to iend[i] do
            Liste[i]:=Liste[i], j;
        end do;
        Liste[i]:=[Liste[i]]:
    end do;
    KartProd:=cartprod([seq(Liste[i], i=1..n)]);
    while not KartProd[finished] do
        Term:=KartProd[nextvalue]();
        delta:=1;
        for i from 1 to n-1 do
            if (op(i, Term) - op(i+1, Term))^2 >= 2 then
                delta:=0;
                break;
            end if;
        end do;
        A:=A+delta;
    end do;
end proc; # Thomas Wieder, Feb 22 2009:
# n -> [a(0),a(1),..,a(n)]
A005773_list := proc(n) local W, m, j, i;
W := proc(i, j, n) option remember;
if min(i, j, n) < 0 or max(i, j) > n then 0
elif n = 0 then if i = 0 and j = 0 then 1 else 0 fi
else W(i-1,j,n-1)+W(i,j-1,n-1)+W(i+1,j-1,n-1) fi end:
[1,seq(add(add(W(i,j,m),i=0..m),j=0..m),m=0..n-1)] end:
A005773_list(27); # Peter Luschny, May 21 2011
A005773 := proc(n)
    option remember;
    if n <= 1 then
        1 ;
    else
        2*n*procname(n-1)+3*(n-2)*procname(n-2) ;
        %/n ;
    end if;
end proc:
seq(A005773(n),n=0..10) ; # R. J. Mathar, Jul 25 2017
  • Mathematica
    CoefficientList[Series[(2x)/(3x-1+Sqrt[1-2x-3x^2]), {x,0,40}], x] (* Harvey P. Dale, Apr 03 2011 *)
a[0]=1; a[n_] := Sum[k/n*Sum[Binomial[n, j]*Binomial[j, 2*j-n-k], {j, 0, n}], {k, 1, n}]; Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Mar 31 2015, after Vladimir Kruchinin *)
A005773[n_] := 2 (-1)^(n+1) JacobiP[n - 1, 3, -n -1/2, -7] / (n^2 + n); A005773[0] := 1; Table[A005773[n], {n, 0, 27}] (* Peter Luschny, May 25 2021 *)
  • PARI
    a(n)=if(n<2,n>=0,(2*n*a(n-1)+3*(n-2)*a(n-2))/n)
  • PARI
    for(n=0, 27, print1(if(n==0, 1, sum(k=0, n-1, (-1)^(n - 1 + k)*binomial(n - 1, k)*binomial(2*k + 1, k + 1))),", ")) \\ Indranil Ghosh, Mar 14 2017
  • PARI
    Vec(1/(1-serreverse(x*(1-x)/(1-x^3) + O(x*x^25)))) \\ Andrew Howroyd, Dec 04 2017
  • Sage
    def da():
    a, b, c, d, n = 0, 1, 1, -1, 1
    yield 1
    yield 1
    while True:
        yield b + (-1)^n*d
        n += 1
        a, b = b, (3*(n-1)*n*a+(2*n-1)*n*b)//((n+1)*(n-1))
        c, d = d, (3*(n-1)*c-(2*n-1)*d)//n
A005773 = da()
print([next(A005773) for  in range(28)]) # _Peter Luschny, May 16 2016
  • Sage
    (2*x/(3*x-1+sqrt(1-2*x-3*x^2))).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, Apr 05 2019

Formula

G.f.: 2*x/(3*x-1+sqrt(1-2*x-3*x^2)). - Len Smiley
Also a(0)=1, a(n) = Sum_{k=0..n-1} M(k)*a(n-k-1), where M(n) are the Motzkin numbers (A001006).
D-finite with recurrence n*a(n) = 2*n*a(n-1) + 3*(n-2)*a(n-2), a(0)=a(1)=1. - Michael Somos, Feb 02 2002
G.f.: 1/2+(1/2)*((1+x)/(1-3*x))^(1/2). Related to Motzkin numbers A001006 by a(n+1) = 3*a(n) - A001006(n-1) [see Yaqubi Lemma 2.6].
a(n) = Sum_{q=0..n} binomial(q, floor(q/2))*binomial(n-1, q) for n > 0. - Emeric Deutsch, Aug 15 2002
From Paul Barry, Jun 22 2004: (Start)
a(n+1) = Sum_{k=0..n} (-1)^(n+k)*C(n, k)*C(2*k+1, k+1).
a(n) = 0^n + Sum_{k=0..n-1} (-1)^(n+k-1)*C(n-1, k)*C(2*k+1, k+1). (End)
a(n+1) = Sum_{k=0..n} (-1)^k*3^(n-k)*binomial(n, k)*A000108(k). - Paul Barry, Jan 27 2005
Starting (1, 2, 5, 13, ...) gives binomial transform of A001405 and inverse binomial transform of A001700. - Gary W. Adamson, Aug 31 2007
Starting (1, 2, 5, 13, 35, 96, ...) gives row sums of triangle A132814. - Gary W. Adamson, Aug 31 2007
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). - Paul Barry, Jan 19 2009
G.f.: 1+x/(1-2*x-x^2/(1-x-x^2/(1-x-x^2/(1-x-x^2/(1-.... (continued fraction). - Paul Barry, Jan 19 2009
a(n) = Sum_{l_1=0..n+1} Sum_{l_2=0..n}...Sum_{l_i=0..n-i}...Sum_{l_n=0..1} delta(l_1,l_2,...,l_i,...,l_n) 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 and delta(l_1,l_2,..., l_i,...,l_n) = 1 otherwise. - Thomas Wieder, Feb 25 2009
INVERT transform of offset Motzkin numbers (A001006): (a(n)){n>=1}=(1,1,2,4,9,21,...). - _David Callan, Aug 27 2009
A005773(n) = ((n+3)*A001006(n+1) + (n-3)*A001006(n)) * (n+2)/(18*n) for n > 0. - Mark van Hoeij, Jul 02 2010
a(n) = Sum_{k=1..n} (k/n * Sum_{j=0..n} binomial(n,j)*binomial(j,2*j-n-k)). - Vladimir Kruchinin, Sep 06 2010
a(0) = 1; a(n+1) = Sum_{t=0..n} n!/((n-t)!*ceiling(t/2)!*floor(t/2)!). - Andrew S. Hays, Feb 02 2011
a(n) = leftmost column term of M^n*V, where M = an infinite quadradiagonal matrix with all 1's in the main, super and subdiagonals, [1,0,0,0,...] in the diagonal starting at position (2,0); and rest zeros. V = vector [1,0,0,0,...]. - Gary W. Adamson, Jun 16 2011
From Gary W. Adamson, Jul 29 2011: (Start)
a(n) = upper left term of M^n, a(n+1) = sum of top row terms of M^n; M = an infinite square production matrix in which the main diagonal is (1,1,0,0,0,...) as follows:
1, 1, 0, 0, 0, 0, ...
1, 1, 1, 0, 0, 0, ...
1, 1, 0, 1, 0, 0, ...
1, 1, 1, 0, 1, 0, ...
1, 1, 1, 1, 0, 1, ...
1, 1, 1, 1, 1, 0, ... (End)
Limit_{n->oo} a(n+1)/a(n) = 3.0 = lim_{n->oo} (1 + 2*cos(Pi/n)). - Gary W. Adamson, Feb 10 2012
a(n) = A025565(n+1) / 2 for n > 0. - Reinhard Zumkeller, Mar 30 2012
With first term deleted: E.g.f.: a(n) = n! * [x^n] exp(x)*(BesselI(0, 2*x) + BesselI(1, 2*x)). - Peter Luschny, Aug 25 2012
G.f.: G(0)/2 + 1/2, where G(k) = 1 + 2*x*(4*k+1)/( (2*k+1)*(1+x) - x*(1+x)*(2*k+1)*(4*k+3)/(x*(4*k+3) + (1+x)*(k+1)/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jun 24 2013
a(n) ~ 3^(n-1/2)/sqrt(Pi*n). - Vaclav Kotesovec, Jul 30 2013
For n > 0, a(n) = (-1)^(n+1) * hypergeom([3/2, 1-n], [2], 4). - Vladimir Reshetnikov, Apr 25 2016
a(n) = GegenbauerC(n-2,-n+1,-1/2) + GegenbauerC(n-1,-n+1,-1/2) for n >= 1. - Peter Luschny, May 12 2016
0 = a(n)*(+9*a(n+1) + 18*a(n+2) - 9*a(n+3)) + a(n+1)*(-6*a(n+1) + 7*a(n+2) - 2*a(n+3)) + a(n+2)*(-2*a(n+2) + a(n+3)) for n >= 0. - Michael Somos, Dec 01 2016
G.f.: 1/(1-x*G(x)) where G(x) is g.f. of A001006. - Andrew Howroyd, Dec 04 2017
a(n) = (-1)^(n + 1)*2*JacobiP(n - 1, 3, -n - 1/2, -7)/(n^2 + n). - Peter Luschny, May 25 2021
a(n+1) = A005043(n) + 2*A005717(n) for n >= 1. - Peter Bala, Feb 11 2022
a(n) = Sum_{k=0..n-1} A064189(n-1,k) for n >= 1. - Alois P. Heinz, Aug 29 2022

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