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.

Previous Showing 11-14 of 14 results.

A064326 Generalized Catalan numbers C(-4; n).

Original entry on oeis.org

1, 1, -3, 25, -251, 2817, -33843, 425769, -5537835, 73865617, -1004862179, 13888533561, -194475377243, 2752994728225, -39333541106835, 566464908534345, -8214515461250955, 119845125957958065, -1757855400878129475, 25906894146115000665, -383443906519878272955
Offset: 0

Views

Author

Wolfdieter Lang, Sep 21 2001

Keywords

Comments

See triangle A064334 with columns m built from C(-m; n), m >= 0, also for Derrida et al. references.

Links

Crossrefs

Cf. A064334.

Programs

  • Magma
    R:=PowerSeriesRing(Rationals(), 30); Coefficients(R!( (9 +Sqrt(1+16*x))/(2*(5-x)) )); // G. C. Greubel, May 03 2019
  • Mathematica
    CoefficientList[Series[(9 +Sqrt[1+16*x])/(2*(5-x)), {x, 0, 30}], x] (* G. C. Greubel, May 03 2019 *)
  • PARI
    my(x='x+O('x^30)); Vec((9 +sqrt(1+16*x))/(2*(5-x))) \\ G. C. Greubel, May 03 2019
  • Sage
    def a(n):
    if n==0: return 1
    return hypergeometric([1-n, n], [-n], -4).simplify()
[a(n) for n in range(20)] # Peter Luschny, Nov 30 2014
  • Sage
    ((9 +sqrt(1+16*x))/(2*(5-x))).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, May 03 2019

Formula

a(n) = Sum_{m=0..n-1} (n-m)*binomial(n-1+m, m)*(-4)^m/n.
a(n) = (1/5)^n*(1 + 4*Sum_{k=0..n-1} C(k)*(-4*5)^k), n >= 1, a(0) = 1; with C(n) = A000108(n) (Catalan).
G.f.: (1+4*x*c(-4*x)/5)/(1-x/5) = 1/(1-x*c(-4*x)) with c(x) g.f. of Catalan numbers A000108.
a(n) = hypergeometric([1-n, n], [-n], -4) for n > 0. - Peter Luschny, Nov 30 2014

A064328 Generalized Catalan numbers C(-6; n).

Original entry on oeis.org

1, 1, -5, 61, -917, 15421, -277733, 5239117, -102188021, 2044131037, -41706059525, 864547613293, -18157111255829, 385517710342909, -8261602828082213, 178459989617336461, -3881680470161846837
Offset: 0

Views

Author

Wolfdieter Lang, Sep 21 2001

Keywords

Comments

See triangle A064334 with columns m built from C(-m; n), m >= 0, also for Derrida et al. references.

Links

Programs

  • Magma
    R:=PowerSeriesRing(Rationals(), 30); Coefficients(R!( (13 +Sqrt(1+24*x))/(2*(7-x)) )); // G. C. Greubel, May 03 2019
  • Mathematica
    CoefficientList[Series[(13 +Sqrt[1+24*x])/(2*(7-x)), {x, 0, 30}], x] (* G. C. Greubel, May 03 2019 *)
  • PARI
    my(x='x+O('x^30)); Vec((13 +sqrt(1+24*x))/(2*(7-x))) \\ G. C. Greubel, May 03 2019
  • Sage
    ((13 +sqrt(1+24*x))/(2*(7-x))).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, May 03 2019

Formula

a(n) = Sum_{m=0..n-1} (n-m)*binomial(n-1+m, m)*(-6)^m/n.
a(n) = (1/7)^n*(1 + 6*Sum_{k=0..n-1} C(k)*(-6*7)^k), n >= 1, a(0) := 1; with C(n)=A000108(n) (Catalan).
G.f.: (1+6*x*c(-6*x)/7)/(1-x/7) = 1/(1-x*c(-6*x)) with c(x) g.f. of Catalan numbers A000108.

A064330 Generalized Catalan numbers C(-8; n).

Original entry on oeis.org

1, 1, -7, 113, -2263, 50721, -1217703, 30622929, -796311415, 21237226625, -577699502407, 15966537989425, -447086291268119, 12656524451911393, -361628025405250023, 10415207118205622673, -302049007052246016183
Offset: 0

Views

Author

Wolfdieter Lang, Sep 21 2001

Keywords

Comments

See triangle A064334 with columns m built from C(-m; n), m >= 0, also for Derrida et al. references.

Links

Programs

  • Magma
    R:=PowerSeriesRing(Rationals(), 30); Coefficients(R!( (17 +Sqrt(1+32*x))/(2*(9-x)) )); // G. C. Greubel, May 03 2019
  • Mathematica
    CoefficientList[Series[(17 +Sqrt[1+32*x])/(2*(9-x)), {x, 0, 30}], x] (* G. C. Greubel, May 03 2019 *)
  • PARI
    my(x='x+O('x^30)); Vec((17 +sqrt(1+32*x))/(2*(9-x))) \\ G. C. Greubel, May 03 2019
  • Sage
    ((17 +sqrt(1+32*x))/(2*(9-x))).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, May 03 2019

Formula

a(n) = Sum_{m=0..n-1} (n-m)*binomial(n-1+m, m)*(-8)^m/n.
a(n) = (1/9)^n*(1 + 8*Sum_{k=0..n-1} C(k)*(-8*9)^k ), n >= 1, a(0) := 1; with C(n)=A000108(n) (Catalan).
G.f.: (1+8*x*c(-8*x)/9)/(1-x/9) = 1/(1-x*c(-8*x)) with c(x) g.f. of Catalan numbers A000108.

A064332 Generalized Catalan numbers C(-10; n).

Original entry on oeis.org

1, 1, -9, 181, -4529, 126861, -3806649, 119653941, -3889122369, 129646443421, -4408213959689, 152290162367301, -5330337257966609, 188617242067457581, -6736489341630231129, 242518500968942706261, -8791448318093732481249
Offset: 0

Views

Author

Wolfdieter Lang, Sep 21 2001

Keywords

Comments

See triangle A064334 with columns m built from C(-m; n), m >= 0, also for Derrida et al. references.

Links

Crossrefs

Cf. A000108, A064334.

Programs

  • Magma
    R:=PowerSeriesRing(Rationals(), 30); Coefficients(R!( (21 +Sqrt(1+40*x))/(2*(11-x)) )); // G. C. Greubel, May 03 2019
  • Mathematica
    CoefficientList[Series[(21 +Sqrt[1+40*x])/(2*(11-x)), {x, 0, 30}], x] (* G. C. Greubel, May 03 2019 *)
  • PARI
    my(x='x+O('x^30)); Vec((21 +sqrt(1+40*x))/(2*(11-x))) \\ G. C. Greubel, May 03 2019
  • Sage
    ((21 +sqrt(1+40*x))/(2*(11-x))).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, May 03 2019

Formula

a(n) = Sum_{m=0..n-1} (n-m)*binomial(n-1+m, m)*(-10)^m/n.
a(n) = (1/11)^n*(1 + 10*Sum_{k=0..n-1} C(k)*(-10*11)^k ), n >= 1, a(0) := 1; with C(n)=A000108(n) (Catalan).
G.f.: (1+10*x*c(-10*x)/11)/(1-x/11) = 1/(1-x*c(-10*x)) with c(x) g.f. of Catalan numbers A000108.
Previous Showing 11-14 of 14 results.

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