A104559 -id:A104559 - cp's OEIS frontend

A104560 Matrix inverse of triangle A104559, read by rows.

1, -1, 1, 1, -2, 1, -2, 5, -4, 1, 6, -16, 15, -6, 1, -30, 81, -79, 36, -9, 1, 204, -552, 543, -256, 72, -12, 1, -1944, 5262, -5184, 2461, -712, 132, -16, 1, 23340, -63180, 62260, -29596, 8615, -1640, 220, -20, 1, -360060, 974670, -960520, 456700, -133091, 25475, -3500, 350, -25, 1, 6692280, -18115800

Offset: 0

Paul D. Hanna, Mar 16 2005

Column 0 is A104561.

			Triangle begins:
1;
-1,1;
1,-2,1;
-2,5,-4,1;
6,-16,15,-6,1;
-30,81,-79,36,-9,1;
204,-552,543,-256,72,-12,1;
-1944,5262,-5184,2461,-712,132,-16,1;
23340,-63180,62260,-29596,8615,-1640,220,-20,1;
-360060,974670,-960520,456700,-133091,25475,-3500,350,-25,1; ...

Cf. A104559, A104561.

{T(n,k)=local(M);M=matrix(n+1,n+1,m,j,if(m>=j, binomial(m-1-(j-1)\2,j\2)*binomial(m-1-j\2,(j-1)\2))); return((M^-1)[n+1,k+1])}

A091964 Number of left factors of peakless Motzkin paths of length n.

Original entry on oeis.org

1, 2, 4, 9, 21, 50, 121, 296, 730, 1812, 4521, 11328, 28485, 71844, 181674, 460443, 1169283, 2974574, 7578937, 19337489, 49401526, 126350742, 323495259, 829033334, 2126454271, 5458711430, 14023219126, 36049991901, 92734505565

Offset: 0

Emeric Deutsch, Mar 13 2004

nonn

Number of paths from (0,0) to the line x=n, consisting of steps u=(1,1), h=(1,0), d=(1,-1), that never go below the x-axis and a u step is never followed by a d step.
a(n) is also the number of peakless Motzkin paths of length n in which the (1,0)-steps at level 0 come in 2 colors. Example: a(4)=21 because, denoting u=(1,1), h=(1,0), and d=(1,-1), we have 2^4 = 16 paths of shape hhhh, 2 paths of shape huhd, 2 paths of shape uhdh, and 1 path of shape uhhd. - Emeric Deutsch, May 03 2011
Equals diagonal sums of triangle A124428. - Paul D. Hanna, Oct 31 2006

			a(2)=4 because we have hh, hu, uh and uu.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Andrei Asinowski, Axel Bacher, Cyril Banderier, Bernhard Gittenberger, Analytic Combinatorics of Lattice Paths with Forbidden Patterns: Enumerative Aspects, in International Conference on Language and Automata Theory and Applications, S. Klein, C. Martín-Vide, D. Shapira (eds), Springer, Cham, pp 195-206, 2018.
Andrei Asinowski, Axel Bacher, Cyril Banderier, Bernhard Gittenberger, Analytic combinatorics of lattice paths with forbidden patterns, the vectorial kernel method, and generating functions for pushdown automata, Laboratoire d'Informatique de Paris Nord (LIPN 2019).
Ivo L. Hofacker, Christian M. Reidys, and Peter F. Stadler, Symmetric circular matchings and RNA folding. Discr. Math., 312:100-112, 2012. See Prop. 5, C_2^{1}(z).
Asamoah Nkwanta, Lattice paths and RNA secondary structures, DIMACS Series in Discrete Math. and Theoretical Computer Science, 34, 1997, 137-147.
Helmut Prodinger, Cornerless, peakless, valleyless Motzkin paths (regular and skew) and applications to bargraphs, arXiv:2501.13645 [math.CO], 2025. See p. 8.

Cf. A004148, A104559, A124428.

[(&+[Binomial(Floor((n+k)/2),k)*Binomial(Floor((n+k+1)/2),k): k in [0..n]]): n in [0..30]]; // G. C. Greubel, Feb 26 2019

CoefficientList[Series[2/(1-3*x+x^2+Sqrt[1-2*x-x^2-2*x^3+x^4]), {x, 0, 20}], x] (* Vaclav Kotesovec, Feb 12 2014 *)

a(n)=sum(k=0,n,binomial(n-k\2,(k+1)\2)*binomial(n-(k+1)\2,k\2)) \\ Paul D. Hanna, Mar 24 2005

a(n)=sum(k=0,n,binomial((n+k)\2,k)*binomial((n+k+1)\2,k)) \\ Paul D. Hanna, Oct 31 2006

[sum(binomial(floor((n+k)/2),k)*binomial(floor((n+k+1)/2),k) for k in (0..n)) for n in (0..30)] # G. C. Greubel, Feb 26 2019

G.f.: 2/(1 - 3*z + z^2 + sqrt(1 - 2*z - z^2 - 2*z^3 + z^4)).
a(n) = Sum_{k=0..n} C(n-floor(k/2), floor((k+1)/2)) * C(n-floor((k+1)/2), floor(k/2)). - Paul D. Hanna, Mar 24 2005
a(n) = Sum_{k=0..n} C(floor((n+k)/2),k)*C(floor((n+k+1)/2),k). - Paul D. Hanna, Oct 31 2006
G.f.: 1/(1-x-x/(1-x^2/(1-x/(1-x^2/(1-x/(1-x^2/(1-... (continued fraction). - Paul Barry, Jun 30 2009
D-finite with recurrence (n+1)*a(n) + 2*(-n-1)*a(n-1) + (-n+1)*a(n-2) + 2*(-n+3)*a(n-3) + (n-3)*a(n-4) = 0. - R. J. Mathar, Nov 24 2012
a(n) ~ (3+sqrt(5))^n / (sqrt(7*sqrt(5)-15) * sqrt(Pi*n) * 2^(n-1/2)). - Vaclav Kotesovec, Feb 12 2014
Equivalently, a(n) ~ phi^(2*n + 2) / (5^(1/4) * sqrt(Pi*n)), where phi = A001622 is the golden ratio. - Vaclav Kotesovec, Dec 08 2021

A104557 Triangle T, read by rows, such that the unsigned columns of the matrix inverse when read downwards equals the rows of T read backwards, with T(n,n)=1 and T(n,n-1) = floor((n+1)/2)*floor((n+2)/2).

Original entry on oeis.org

1, 1, 1, 2, 2, 1, 6, 6, 4, 1, 24, 24, 18, 6, 1, 120, 120, 96, 36, 9, 1, 720, 720, 600, 240, 72, 12, 1, 5040, 5040, 4320, 1800, 600, 120, 16, 1, 40320, 40320, 35280, 15120, 5400, 1200, 200, 20, 1, 362880, 362880, 322560, 141120, 52920, 12600, 2400, 300, 25, 1

Offset: 0

Paul D. Hanna, Mar 16 2005

Matrix inverse is A104558. Row sums form A102038. See A104559 for further formulas, where A104559(n,k) = T(n,k)/(n-k)!.

			Rows of T begin:
      1;
      1,     1;
      2,     2,     1;
      6,     6,     4,     1;
     24,    24,    18,     6,    1;
    120,   120,    96,    36,    9,    1;
    720,   720,   600,   240,   72,   12,   1;
   5040,  5040,  4320,  1800,  600,  120,  16,  1;
  40320, 40320, 35280, 15120, 5400, 1200, 200, 20, 1; ...
The matrix inverse A104558 begins:
   1;
  -1,  1;
   0, -2,  1;
   0,  2, -4,   1;
   0,  0,  6,  -6,   1;
   0,  0, -6,  18,  -9,   1;
   0,  0,  0, -24,  36, -12,   1;
   0,  0,  0,  24, -96,  72, -16, 1; ...

Cf. A104558, A104559, A102038.

T(n,k)=(n-k)!*binomial(n-(k\2),(k+1)\2)*binomial(n-((k+1)\2),k\2)

Formula: T(n,k) = (n-k)!*C(n-floor(k/2), floor((k+1)/2))*C(n-floor((k+1)/2), floor(k/2)).
Recurrence: T(n,k) = n*T(n-1,k) + T(n-2,k-2) for n >= k >= 2, with T(0,0) = T(1,0) = T(1,1) = 1.
T(n,0) = n!.
T(n,k) = T(n-1,k-1) + floor((k+2)/2)*T(n,k+1), T(0,0)=1, T(n,k)=0 for k > n or for k < 0. - Philippe Deléham, Dec 18 2006

A104561 Column 0 of triangle A104560.

Original entry on oeis.org

1, -1, 1, -2, 6, -30, 204, -1944, 23340, -360060, 6692280, -151908120, 4032593040, -126425330640, 4537069829880, -187762669281600, 8751745668796200, -462088269954518760, 27132285729049971120, -1779812144439507990000, 128512316060872730854560

Offset: 0

Paul D. Hanna, Mar 16 2005

sign

A104560 equals the matrix inverse of A104559, where A104559(n,k) = C(n-[k/2],[(k+1)/2])*C(n-[(k+1)/2],[k/2]).

Cf. A104559, A104560.

{a(n)=local(M);M=matrix(n+1,n+1,m,j,if(m>=j, binomial(m-1-(j-1)\2,j\2)*binomial(m-1-j\2,(j-1)\2))); return((M^-1)[n+1,1])}

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A104560 Matrix inverse of triangle A104559, read by rows.

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A091964 Number of left factors of peakless Motzkin paths of length n.

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A104557 Triangle T, read by rows, such that the unsigned columns of the matrix inverse when read downwards equals the rows of T read backwards, with T(n,n)=1 and T(n,n-1) = floor((n+1)/2)*floor((n+2)/2).

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A104561 Column 0 of triangle A104560.

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