A091561 -id:A091561 - cp's OEIS frontend

A025265 a(n) = a(1)a(n-1) + a(2)a(n-2) + ... + a(n-1)*a(1) for n >= 4.

1, 0, 1, 2, 4, 9, 22, 56, 146, 388, 1048, 2869, 7942, 22192, 62510, 177308, 506008, 1451866, 4185788, 12119696, 35227748, 102753800, 300672368, 882373261, 2596389190, 7658677856, 22642421206, 67081765932, 199128719896, 592179010350, 1764044315540

Offset: 1

Clark Kimberling

nonn

With offset 0, a(n) is the number of 021-avoiding ascent sequences of length n with no isolated 0's. For example, a(4)=4 counts 0000, 0001, 0011, 0012. - David Callan, Nov 13 2019

G. C. Greubel, Table of n, a(n) for n = 1..1000
Jean-Luc Baril, Rigoberto Flórez, and José L. Ramírez, Counting symmetric and asymmetric peaks in motzkin paths with air pockets, Univ. Bourgogne (France, 2023).
Paul Barry, Jacobsthal Decompositions of Pascal's Triangle, Ternary Trees, and Alternating Sign Matrices, Journal of Integer Sequences, 19, 2016, #16.3.5.
David Callan, On Ascent, Repetition and Descent Sequences, arXiv:1911.02209 [math.CO], 2019.

Cf. A025247, A091561.

nmax = 30; aa = ConstantArray[0,nmax]; aa[[1]] = 1; aa[[2]] = 0; aa[[3]] = 1; Do[aa[[n]] = Sum[aa[[k]] * aa[[n-k]],{k,1,n-1}],{n,4,nmax}]; aa (* Vaclav Kotesovec, Jan 25 2015 *)
CoefficientList[Series[(1-Sqrt[1-4x+4x^2-4x^3])/2,{x,0,40}],x] (* Harvey P. Dale, Jun 02 2017 *)

a(n):=sum((binomial(2*k,k)*(sum(binomial(j,n-k-j-1)*binomial(k+1,j),j,0,k+1))*(-1)^(-n+k+1))/(k+1),k,0,n); /* Vladimir Kruchinin, May 10 2018  */

a(n)=polcoeff((1-sqrt(1-4*x+4*x^2-4*x^3+x*O(x^n)))/2,n)

a(n)=if(n<1,0,polcoeff(subst(serreverse(x-x^2+x*O(x^n)),x,x-x^2+x^3),n))

a(n+2) = A091561(n).
G.f.: (1-sqrt(1-4*x+4*x^2-4*x^3))/2. - Michael Somos, Jun 08 2000
G.f. A(x) satisfies 0=f(x, A(x)) where f(x, y)=(x-x^2+x^3)-(y-y^2). - Michael Somos, May 26 2005
D-finite with recurrence n*a(n) +2*(3-2*n)*a(n-1) +4*(n-3)*a(n-2)+ 2*(9-2*n)*a(n-3)=0. - R. J. Mathar, Aug 14 2012
a(n) = Sum_{k=0..n} C(k)*Sum_{j=0..k+1} binomial(j,n-k-j-1)*binomial(k+1,j)*(-1)^(-n+k-1), where C(k) is Catalan numbers (A000108)  - Vladimir Kruchinin, May 10 2018

A152225 Number of Dyck paths of semilength n with no peaks at height 0 (mod 3) and no valleys at height 2 (mod 3).

Original entry on oeis.org

1, 1, 2, 4, 9, 22, 56, 146, 388, 1048, 2869, 7942, 22192, 62510, 177308, 506008, 1451866, 4185788, 12119696, 35227748, 102753800, 300672368, 882373261, 2596389190, 7658677856, 22642421206, 67081765932, 199128719896, 592179010350, 1764044315540, 5263275015120

Offset: 0

Majun (majun(AT)math.sinica.edu.tw), Nov 29 2008

nonn

The antidiagonal sums of A091894 equal this sequence. - Johannes W. Meijer, Sep 13 2012

Robert Israel, Table of n, a(n) for n = 0..2025
Jean-Luc Baril, Sergey Kirgizov, and Armen Petrossian, Forests and pattern-avoiding permutations modulo pure descents, Permutation Patterns 2017, Reykjavik University, Iceland, June 26-30, 2017. See Section 5.
Jean-Luc Baril and José Luis Ramírez, Descent distribution on Catalan words avoiding ordered pairs of Relations, arXiv:2302.12741 [math.CO], 2023.
Paul Barry, Jacobsthal Decompositions of Pascal's Triangle, Ternary Trees, and Alternating Sign Matrices, Journal of Integer Sequences, 19, 2016, #16.3.5.
Paul Barry, Riordan Pseudo-Involutions, Continued Fractions and Somos 4 Sequences, arXiv:1807.05794 [math.CO], 2018.
Shu-Chung Liu, Jun Ma, and Yeong-Nan Yeh, Dyck Paths with Peak- and Valley-Avoiding Sets, Stud. Appl Math. 121 (3) (2008) 263-289.

Cf. A091561, A025265, A025247. - R. J. Mathar, Dec 03 2008

f:= gfun:-rectoproc({(n+2)*a(n) - 2*(2*n+1)*a(n-1) + 4*(n-1)*a(n-2) + 2*(5-2*n)*a(n-3)=0,a(0)=1,a(1)=1,a(2)=2,a(3)=4},a(n),remember):
map(f, [$0..40]); # Robert Israel, Jan 09 2018

CoefficientList[Series[(1 - 2 x + 2 x^2 - Sqrt[1 - 4 x + 4 x^2 - 4 x^3])/(2 x^2), {x, 0, 30}], x] (* Michael De Vlieger, Jan 09 2018 *)

G.f.: (1 - 2*x + 2*x^2 - sqrt(1 - 4*x + 4*x^2 - 4*x^3))/(2*x^2).
Conjecture: (n+2)*a(n) - 2*(2*n+1)*a(n-1) + 4*(n-1)*a(n-2) + 2*(5-2*n)*a(n-3)=0. - R. J. Mathar, Aug 14 2012
This conjecture follows from the differential equation (4*x^4-4*x^3+4*x^2-x)*y' + (2*x^3-4*x^2+6*x-2)*y - 2*x^3+2*x^2-3*x+2=0 satisfied by the g.f. - Robert Israel, Jan 09 2018

Edited by Emeric Deutsch, Dec 20 2008

A273896 Triangle read by rows: T(n,k) is the number of bargraphs of semiperimeter n having k UHU configurations, where U=(0,1), H(1,0); (n>=2, k>=0).

Original entry on oeis.org

1, 2, 4, 1, 9, 4, 22, 12, 1, 56, 35, 6, 146, 104, 24, 1, 388, 312, 86, 8, 1048, 938, 300, 40, 1, 2869, 2824, 1032, 170, 10, 7942, 8520, 3502, 680, 60, 1, 22192, 25763, 11748, 2632, 295, 12, 62510, 78064, 39072, 9926, 1330, 84, 1, 177308, 236976, 129100, 36640, 5712, 469, 14, 506008, 720574, 424344, 132960, 23660, 2352, 112, 1

Offset: 2

Emeric Deutsch, Jun 02 2016

Sum of entries in row n = A082582(n).
T(n,0) = A091561(n-1).
Sum(k*T(n,k), k>=0) = A273714(n-1). This implies that the number of UHUs in all bargraphs of semiperimeter n is equal to the number of doublerises in all bargraphs of semiperimeter n-1.

			Row 4 is [4,1] because the 5 (=A082582(4)) bargraphs of semiperimeter 4 correspond to the compositions [1,1,1], [1,2], [2,1], [2,2], [3] and the corresponding drawings show that they have 0,1,0,0,0 UHU's.
Triangle starts
1;
2;
4,1;
9,4;
22,12,1;
56,35,6.

Alois P. Heinz, Rows n = 2..200, flattened
M. Bousquet-Mélou and A. Rechnitzer, The site-perimeter of bargraphs, Adv. in Appl. Math. 31 (2003), 86-112.
Emeric Deutsch, S Elizalde, Statistics on bargraphs viewed as cornerless Motzkin paths, arXiv preprint arXiv:1609.00088, 2016

Cf. A091561, A082582, A273714.

eq := z*G^2-(1-2*z-t*z^2)*G+z^2 = 0: G := RootOf(eq, G): Gser := simplify(series(G, z = 0, 20)): for n from 2 to 18 do P[n] := sort(coeff(Gser, z, n)) end do: for n from 2 to 18 do seq(coeff(P[n], t, j), j = 0 .. degree(P[n])) end do; # yields sequence in triangular form
# second Maple program:
b:= proc(n, y, t, h) option remember; expand(
      `if`(n=0, (1-t), `if`(t<0, 0, b(n-1, y+1, 1, 0)*z^h)+
      `if`(t>0 or y<2, 0, b(n, y-1, -1, 0))+
      `if`(y<1, 0, b(n-1, y, 0, `if`(t>0, 1, 0)))))
    end:
T:= n-> (p-> seq(coeff(p, z, i), i=0..degree(p)))(b(n, 0$3)):
seq(T(n), n=2..22); # Alois P. Heinz, Jun 06 2016

b[n_, y_, t_, h_] := b[n, y, t, h] = Expand[If[n==0, 1-t, If[t<0, 0, b[n-1, y+1, 1, 0]*z^h] + If[t>0 || y<2, 0, b[n, y-1, -1, 0]] + If[y<1, 0, b[n-1, y, 0, If[t>0, 1, 0]]]]]; T[n_] := Function[p, Table[Coefficient[p, z, i], {i, 0, Exponent[p, z]}]][b[n, 0, 0, 0]]; Table[T[n], {n, 2, 22}] // Flatten (* Jean-François Alcover, Dec 02 2016 after Alois P. Heinz *)

G.f.: G=G(t,z), where t marks number of UHU's and z marks semiperimeter, satisfies zG^2-(1-2z-tz^2)G+z^2 = 0.

cp's OEIS Frontend

A025265 a(n) = a(1)a(n-1) + a(2)a(n-2) + ... + a(n-1)*a(1) for n >= 4.

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A152225 Number of Dyck paths of semilength n with no peaks at height 0 (mod 3) and no valleys at height 2 (mod 3).

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A273896 Triangle read by rows: T(n,k) is the number of bargraphs of semiperimeter n having k UHU configurations, where U=(0,1), H(1,0); (n>=2, k>=0).

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cp's OEIS Frontend

A025265 a(n) = a(1)*a(n-1) + a(2)*a(n-2) + ... + a(n-1)*a(1) for n >= 4.

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Author

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Crossrefs

Programs

Mathematica

Maxima

PARI

PARI

Formula

A152225 Number of Dyck paths of semilength n with no peaks at height 0 (mod 3) and no valleys at height 2 (mod 3).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

Formula

Extensions

A273896 Triangle read by rows: T(n,k) is the number of bargraphs of semiperimeter n having k UHU configurations, where U=(0,1), H(1,0); (n>=2, k>=0).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A025265 a(n) = a(1)a(n-1) + a(2)a(n-2) + ... + a(n-1)*a(1) for n >= 4.