cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-5 of 5 results.

A252354 Number of Motzkin paths of length n with no level steps at height 2.

Original entry on oeis.org

1, 1, 2, 4, 9, 20, 46, 106, 248, 584, 1389, 3329, 8047, 19607, 48167, 119287, 297829, 749632, 1902044, 4864553, 12538933, 32568528, 85224251, 224618900, 596106393, 1592429464, 4280667705, 11575188106, 31474407317, 86029586086, 236292044931, 651952466845
Offset: 0

Views

Author

José Luis Ramírez Ramírez, Apr 20 2015

Keywords

Links

Crossrefs

Cf. A005043, A217312.

Programs

  • Mathematica
    CoefficientList[Series[1/(1-x-x^2(1/(1-x-x^2*(1+x-Sqrt[1-2*x-3*x^2])/(2*x*(1+x))))), {x, 0, 20}], x] (* Vaclav Kotesovec, Apr 21 2015 *)
  • PARI
    x='x + O('x^50); Vec(1/(1-x-x^2*(1/(1-x-x^2*(1+x-sqrt(1-2*x-3*x^2))/(2*x*(1+x)))))) \\ G. C. Greubel, Feb 14 2017

Formula

a(n) = a(n-1) + Sum_{j=0..n-2} A217312(j)*a(n-j).
G.f: 1/(1-x-x^2(1/(1-x-x^2*R(x)))), where R(x) is the g.f. of Riordan numbers (A005043).
a(n) ~ 3^(n+3/2) / (32*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Apr 21 2015
Conjecture: (-n+3)*a(n) +3*(2*n-7)*a(n-1) +(-7*n+24)*a(n-2) +2*(-7*n+36)*a(n-3) +2*(11*n-51)*a(n-4) +3*(3*n-23)*a(n-5) +(-10*n+63)*a(n-6) +3*(n-6)*a(n-7)=0. - R. J. Mathar, Sep 24 2016

A253831 Number of 2-Motzkin paths with no level steps at height 1.

Original entry on oeis.org

1, 2, 5, 12, 30, 76, 197, 522, 1418, 3956, 11354, 33554, 102104, 319608, 1027237, 3381714, 11371366, 38946892, 135505958, 477781296, 1703671604, 6132978608, 22256615602, 81327116484, 298938112816, 1104473254912, 4098996843500, 15272792557230, 57106723430892, 214202598271360, 805743355591301
Offset: 0

Views

Author

José Luis Ramírez Ramírez, Apr 20 2015

Keywords

Comments

For n=3 we have 12 paths: H(1)H(1)H(1), H(1)H(1)H(2), H(1)H(2)H(1), H(1)H(2)H(2), H(2)H(1)H(1), H(2)H(1)H(2), H(2)H(2)H(1), H(2)H(2)H(2), UDH(1), UDH(2), H(1)UD, H(2)UD.

Links

Crossrefs

Cf. A000957, A217312, A257363.

Programs

  • Maple
    rec:= (54+36*n)*a(n)+(-3+7*n)*a(n+1)+(-60-36*n)*a(n+2)+(36+16*n)*a(n+3)+(-6-2*n)*a(n+4) = 0:
f:= gfun:-rectoproc({rec,seq(a(i)=[1,2,5,12][i+1],i=0..3)},a(n),remember):
seq(f(n),n=0..100); # Robert Israel, Apr 29 2015
  • Mathematica
    CoefficientList[Series[1/(1-2*x-x*((1-Sqrt[1-4*x])/(3-Sqrt[1-4*x]))), {x, 0, 20}], x] (* Vaclav Kotesovec, Apr 21 2015 *)
  • Maxima
    a(n):=sum(sum(((sum((k+1)*binomial(k+m,k+1)*binomial(2*j-k+m-1,j-k)*(-1)^(k),k,0,j))*2^(n-j-2*m)*binomial(n-m-j,m))/(j+m),j,0,n-2*m),m,1,n/2)+2^n; /* Vladimir Kruchinin, Mar 11 2016 */

Formula

G.f.: 1/(1-2*x-x*F(x)), where F(x) is the g.f. of Fine numbers A000957.
G.f.: 2*(2+x)/(4-7*x-6*x^2+x*sqrt(1-4*x)).
a(n) ~ 4^(n+1) / (25*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Apr 21 2015
(54+36*n)*a(n)+(-3+7*n)*a(n+1)+(-60-36*n)*a(n+2)+(36+16*n)*a(n+3)+(-6-2*n)*a(n+4) = 0. - Robert Israel, Apr 29 2015
a(n) = Sum_{m=1..n/2}(Sum_{j=0..n-2*m}(((Sum_{k=0..j}((k+1)*binomial(k+m,k+1)*binomial(2*j-k+m-1,j-k)*(-1)^(k)))*2^(n-j-2*m)*binomial(n-m-j,m))/(j+m)))+2^n. - Vladimir Kruchinin, Mar 11 2016

A257363 Number of 3-Motzkin paths with no level steps at height 1.

Original entry on oeis.org

1, 3, 10, 33, 110, 369, 1247, 4248, 14603, 50724, 178314, 635526, 2300829, 8477382, 31842897, 122103276, 478372886, 1915188093, 7831613468, 32674683984, 138871668314, 600140517762, 2631926843602, 11690520554421, 52498671870181, 237966449687118, 1087246253873875, 5001141997115010, 23137102115963262
Offset: 0

Views

Author

José Luis Ramírez Ramírez, Apr 20 2015

Keywords

Comments

For n=2 we have 10 paths: H(1)H(1), H(1)H(2), H(2)H(1), H(2)H(2), H(1)H(3), H(3)H(1), H(3)H(3), H(2)H(3), H(3)H(2), UD.

Links

Crossrefs

Cf. A117641, A217312, A253831, A059231.

Programs

  • Maple
    rec:= (95+95*n)*a(n)+(-180-9*n)*a(n+1)+(-329-197*n)*a(n+2)+(369+144*n)*a(n+3)+(-117-36*n)*a(4+n)+(12+3*n)*a(n+5):
f:= gfun:-rectoproc({rec,a(0)=1,a(1)=3,a(2)=10,a(3)=33,a(4)=110},a(n),remember):
seq(f(n),n=0..100); # Robert Israel, Apr 28 2015
  • Mathematica
    CoefficientList[Series[2*(3+x)/(6-17*x-9*x^2+x*Sqrt[1-6*x+5*x^2]), {x, 0, 20}], x] (* Vaclav Kotesovec, Apr 21 2015 *)

Formula

G.f.: 1/(1-3*x-x*F(x)), where F(x) is the g.f. of the sequence A117641.
G.f.: 2*(3+x)/(6-17*x-9*x^2+x*sqrt(1-6*x+5*x^2)).
a(n) ~ 5^(n+3/2)/(98*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Apr 21 2015
From Robert Israel, Apr 28 2015 (Start):
G.f.: (6-x*sqrt(1-6*x+5*x^2)-17*x-9*x^2)/(6-36*x+42*x^2+38*x^3).
3*(-n+1)*a(n) +9*(4*n-7)*a(n-1) +9*(-16*n+39)*a(n-2) +(197*n-656)*a(n-3) +9*(n+15)*a(n-4) +95*(-n+4)*a(n-5)=0. (End)

A257386 Number of Motzkin paths of length n with no level steps at height 3.

Original entry on oeis.org

1, 1, 2, 4, 9, 21, 51, 126, 316, 799, 2034, 5202, 13357, 34407, 88888, 230237, 597829, 1555962, 4058944, 10612102, 27807135, 73025751, 192204957, 507025163, 1340545113, 3552492126
Offset: 0

Views

Author

José Luis Ramírez Ramírez, Apr 21 2015

Keywords

Links

Crossrefs

Cf. A005043, A217312, A252354.

Programs

  • Mathematica
    CoefficientList[Series[1/(1-x-x^2*(1/(1-x-x^2*(1/(1-x-x^2*(1+x-Sqrt[1-2*x-3*x^2])/(2*x*(1+x))))))), {x, 0, 20}], x] (* Vaclav Kotesovec, Apr 24 2015 *)
  • PARI
    x='x+O('x^50); Vec(1/(1-x-x^2*(1/(1-x-x^2*(1/(1-x-x^2*(1+x-sqrt(1-2*x-3*x^2))/(2*x*(1+x)))))))) \\ G. C. Greubel, Apr 08 2017

Formula

a(n) = a(n-1) + Sum_{j=0..n-2} A252354(j)*a(n-j).
G.f: 1/(1-x-x^2*(1/(1-x-x^2*(1/(1-x-x^2*R(x)))))), where R(x) is the g.f. of Riordan numbers (A005043).
a(n) ~ 3^(n+3/2)/(50*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Apr 24 2015

A257387 Number of Motzkin paths of length n with no level steps at height 4.

Original entry on oeis.org

1, 1, 2, 4, 9, 21, 51, 127, 323, 834, 2179, 5743, 15238, 40637, 108800, 292200, 786703, 2122387, 5735596, 15522682, 42064028, 114117541, 309918698, 842489130, 2292332265, 6242655886
Offset: 0

Views

Author

José Luis Ramírez Ramírez, Apr 21 2015

Keywords

Links

Crossrefs

Cf. A005043, A217312, A252354, A257386.

Programs

  • Mathematica
    CoefficientList[Series[1/(1-x-x^2*(1/(1-x-x^2*(1/(1-x-x^2*(1/(1-x-x^2*(1+x-Sqrt[1-2*x-3*x^2])/(2*x*(1+x))))))))), {x, 0, 20}], x] (* Vaclav Kotesovec, Apr 24 2015 *)
  • PARI
    x='x+O('x^50); Vec(1/(1-x-x^2*(1/(1-x-x^2*(1/(1-x-x^2*(1/(1-x-x^2*(1+x-sqrt(1-2*x-3*x^2))/(2*x*(1+x)))))))))) \\ G. C. Greubel, Jun 03 2017

Formula

a(n) = a(n-1) + Sum_{j=0..n-2} A257386(j)*a(n-j).
G.f: 1/(1-x-x^2*(1/(1-x-x^2*(1/(1-x-x^2*(1/(1-x-x^2*R(x)))))))), where R(x) is the g.f. of Riordan numbers (A005043).
a(n) ~ 3^(n+1/2)/(24*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Apr 24 2015
Showing 1-5 of 5 results.

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