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 1-3 of 3 results.

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

Original entry on oeis.org

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

Views

Author

Clark Kimberling

Keywords

Comments

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

Links

Crossrefs

Cf. A025247, A091561.

Programs

  • Mathematica
    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 *)
  • Maxima
    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  */
  • PARI
    a(n)=polcoeff((1-sqrt(1-4*x+4*x^2-4*x^3+x*O(x^n)))/2,n)
  • PARI
    a(n)=if(n<1,0,polcoeff(subst(serreverse(x-x^2+x*O(x^n)),x,x-x^2+x^3),n))

Formula

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

A091561 Expansion of (1-2x-sqrt(1-4x+4x^2-4x^3))/(2x^2).

Original entry on oeis.org

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
Offset: 1

Views

Author

Michael Somos, Jan 20 2004

Keywords

Links

Crossrefs

Cf. A025247, A025265.

Programs

  • Mathematica
    CoefficientList[Series[(1-2x-Sqrt[1-4x+4x^2-4x^3])/(2x^2),{x,0,30}],x] (* Harvey P. Dale, Jan 31 2015 *)
  • PARI
    a(n)=polcoeff((1-2*x-sqrt(1-4*x+4*x^2-4*x^3+x^3*O(x^n)))/2,n+2)

Formula

G.f.: (1-2x-sqrt(1-4x+4x^2-4x^3))/(2x^2).
a(n) = 2*a(n-1)+a(1)*a(n-3)+a(2)*a(n-4)+...+a(n-3)*a(1) for n>1.
Series reversion of g.f. A(x) is -A(-x).
G.f. A(x) satisfies 0=f(x, A(x)) where f(x, y)=(xy)^2+2(xy)-(y-x).
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

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

Views

Author

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

Keywords

Comments

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

Links

Crossrefs

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

Programs

  • Maple
    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
  • Mathematica
    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 *)

Formula

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

Extensions

Edited by Emeric Deutsch, Dec 20 2008
Showing 1-3 of 3 results.

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