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.

A191649 Number of lattice paths from (0,0) to (n,n) using steps (0,1), (1,0), (1,1), (2,2).

Original entry on oeis.org

1, 3, 14, 71, 379, 2082, 11651, 66051, 378064, 2180037, 12644861, 73695358, 431209313, 2531556197, 14904832196, 87970766447, 520337606401, 3083584244460, 18304476242735, 108820740004749, 647817646760368, 3861215365595659, 23039691494489015, 137615812845579390
Offset: 0

Views

Author

Joerg Arndt, Jun 30 2011

Keywords

Links

Crossrefs

Cf. A001850, A026641, A036355, A137644, A192364, A192365, A192369, A191354.

Programs

  • Magma
    R:=PowerSeriesRing(Rationals(), 30); Coefficients(R!( 1/Sqrt(x^4+2*x^3-x^2-6*x+1) )); // G. C. Greubel, Apr 29 2019
  • Mathematica
    CoefficientList[Series[1/Sqrt[x^4 + 2 x^3 - x^2 - 6 x + 1], {x, 0, 23}], x] (* Michael De Vlieger, Oct 08 2016 *)
  • PARI
    /* same as in A092566 but use */
steps=[[0,1], [1,0], [1,1], [2,2]];
/* Joerg Arndt, Jun 30 2011 */
  • PARI
    my(x='x+O('x^30)); Vec(1/sqrt(x^4+2*x^3-x^2-6*x+1)) \\ G. C. Greubel, Apr 29 2019
  • Sage
    (1/sqrt(x^4+2*x^3-x^2-6*x+1)).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, Apr 29 2019

Formula

G.f.: 1/sqrt(x^4 +2*x^3 -x^2 -6*x +1). - Mark van Hoeij, Apr 17 2013
D-finite with recurrence: n*a(n) +3*(-2*n+1)*a(n-1) +(-n+1)*a(n-2) +(2*n-3)*a(n-3) +(n-2)*a(n-4)=0. - R. J. Mathar, Oct 08 2016

A192417 Number of lattice paths from (0,0) to (n,n) using steps (0,1), (1,0), (2,2), (3,3).

Original entry on oeis.org

1, 2, 7, 27, 107, 436, 1810, 7609, 32288, 138009, 593311, 2562725, 11112720, 48347332, 210936119, 922550622, 4043488129, 17755735241, 78099099877, 344033901804, 1517535718392, 6701979806379, 29630948706756, 131136723532257, 580901892464599, 2575423975663301
Offset: 0

Views

Author

Joerg Arndt, Jun 30 2011

Keywords

Links

Crossrefs

Cf. A001850, A026641, A036355, A137644, A192364, A192365, A192369, A191354.

Programs

  • Magma
    R:=PowerSeriesRing(Rationals(), 30); Coefficients(R!( 1/Sqrt(x^6+2*x^5+x^4-2*x^3-2*x^2-4*x+1) )); // G. C. Greubel, Apr 29 2019
  • Mathematica
    CoefficientList[Series[1/Sqrt[x^6+2x^5+x^4-2x^3-2x^2-4x+1], {x, 0, 25}], x] (* Michael De Vlieger, Oct 08 2016 *)
  • PARI
    /* same as in A092566 but use */
steps=[[0,1], [1,0], [2,2], [3,3]];
/* Joerg Arndt, Jun 30 2011 */
  • PARI
    my(x='x+O('x^30)); Vec(1/sqrt(x^6+2*x^5+x^4-2*x^3-2*x^2-4*x+1)) \\ G. C. Greubel, Apr 29 2019
  • Sage
    (1/sqrt(x^6+2*x^5+x^4-2*x^3-2*x^2-4*x+1)).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, Apr 29 2019

Formula

G.f.: 1/sqrt(x^6+2*x^5+x^4-2*x^3-2*x^2-4*x+1). - Mark van Hoeij, Apr 17 2013
D-finite with recurrence: n*a(n) +2*(-2*n+1)*a(n-1) +2*(-n+1)*a(n-2) +(-2*n+3)*a(n-3) +(n-2)*a(n-4) +(2*n-5)*a(n-5) +(n-3)*a(n-6)=0. - R. J. Mathar, Oct 08 2016

A191678 Number of lattice paths from (0,0) to (n,n) using steps (1,0), (1,1), (0,2), (2,2).

Original entry on oeis.org

1, 1, 5, 15, 62, 233, 937, 3729, 15121, 61492, 251942, 1036215, 4279754, 17731181, 73670725, 306823695, 1280574706, 5354602495, 22426876445, 94070238840, 395106054632, 1661489413472, 6994494531010, 29474635716345, 124319047552309, 524797934104312, 2217091297558466, 9373180869094923
Offset: 0

Views

Author

Joerg Arndt, Jun 30 2011

Keywords

Crossrefs

Cf. A001850, A026641, A036355, A137644, A192364, A192365, A192369, A191354.

Programs

  • Maple
    P := (4*x^6+12*x^5-20*x^3+27*x^2+12*x-4)*A^3-(3*x^2+3*x-3)*A+1;
Q := eval(P, A=A+1):
series(RootOf(Q,A)+1, x=0, 30);  # Mark van Hoeij, Apr 17 2013
  • PARI
    /* same as in A092566 but use */
steps=[[1,0], [1,1], [0,2], [2,2]];
/* Joerg Arndt, Jun 30 2011 */

Formula

G.f.: A(x) where (4*x^6+12*x^5-20*x^3+27*x^2+12*x-4)*A(x)^3-(3*x^2+3*x-3)*A(x)+1 = 0. - Mark van Hoeij, Apr 17 2013
Showing 1-3 of 3 results.

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