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 1-2 of 2 results.

A069723 a(n) = 2^(n-1)*binomial(2*n-3, n-1).

Original entry on oeis.org

1, 2, 12, 80, 560, 4032, 29568, 219648, 1647360, 12446720, 94595072, 722362368, 5538111488, 42600857600, 328635187200, 2541445447680, 19696202219520, 152935217233920, 1189496134041600, 9265548833587200, 72271280901980160, 564404288948797440, 4412615349963325440
Offset: 1

Views

Author

Valery A. Liskovets, Apr 07 2002

Keywords

Comments

Number of rooted unicursal planar maps with n edges and two vertices of valency 1 (unicursal means that exactly two vertices are of odd valency; there is an Eulerian path).

Links

Crossrefs

Main diagonal of array A082137.
Cf. A069720, A069721, A069722, A082143, A082144, A082145, A088218.

Programs

  • Maple
    Z:=(1-sqrt(1-z))*8^n/sqrt(1-z)/2: Zser:=series(Z, z=0, 33): seq(coeff(Zser, z, n), n=0..20); # Zerinvary Lajos, Jan 16 2007
  • Mathematica
    Table[2^(n - 1) * Binomial[2*n - 3, n - 1], {n, 1,50}] (* G. C. Greubel, Jan 15 2017 *)
  • Sage
    # Assuming offset 0:
A069723  = lambda n: (rising_factorial(n, n)/factorial(n)) << n
[A069723(n) for n in (0..20)] # Peter Luschny, Nov 30 2014

Formula

a(n) = A069722(n)/2, n>1.
G.f.: 4*x/(sqrt(1-8*x) * (1-sqrt(1-8*x))). - Paul Barry, Sep 06 2004
With offset 0: a(n) = (0^n + 2^n*binomial(2n, n))/2. - Paul Barry, Sep 24 2004
D-finite with recurrence (-n+1)*a(n) + 4*(2*n-3)*a(n-1) = 0. - R. J. Mathar, Dec 03 2012
With offset 0: a(n) = 2^n*rf(n,n)/n! = 2^n*A088218(n), where rf denotes the rising factorial. - Peter Luschny, Nov 30 2014
a(n) = Sum_{k=0..n} binomial(n+k-1,k)*binomial(2*n-1, n-k). - Vladimir Kruchinin, Nov 11 2016
a(n) ~ 2^(3*n-4)/sqrt(Pi*n). - Ilya Gutkovskiy, Nov 11 2016
From Amiram Eldar, Jan 16 2024: (Start)
Sum_{n>=1} 1/a(n) = 9/7 + 16*arcsin(1/(2*sqrt(2)))/(7*sqrt(7)).
Sum_{n>=1} (-1)^(n+1)/a(n) = 7/9 - 8*log(2)/27. (End)

A069722 Number of rooted unicursal planar maps with n edges and exactly one vertex of valency 1 (unicursal means that exactly two vertices are of odd valency; there is an Eulerian path).

Original entry on oeis.org

0, 4, 24, 160, 1120, 8064, 59136, 439296, 3294720, 24893440, 189190144, 1444724736, 11076222976, 85201715200, 657270374400, 5082890895360, 39392404439040, 305870434467840, 2378992268083200, 18531097667174400, 144542561803960320, 1128808577897594880
Offset: 1

Views

Author

Valery A. Liskovets, Apr 07 2002

Keywords

Links

Crossrefs

Cf. A059304, A069720, A069721, A069723, A089156.

Programs

  • Magma
    [0] cat[2^(n-1)*Binomial(2*n-2, n-1): n in [2..20]]; // Vincenzo Librandi, Nov 17 2011
  • Maple
    Z:=(1-sqrt(1-z))*8^n/sqrt(1-z): Zser:=series(Z, z=0, 32): seq(coeff(Zser, z, n), n=0..19); # Zerinvary Lajos, Jan 01 2007
  • Mathematica
    Join[{0},Table[2^(n-1) Binomial[2n-2,n-1],{n,2,20}]] (* Harvey P. Dale, Nov 16 2011 *)

Formula

a(n) = 2^(n-1)*binomial(2n-2, n-1), n>1.
a(n) = 2*A069723(n), n>1.
G.f. for a(n)^2: 1/AGM(1, (1-64*x)^(1/2)). - Benoit Cloitre, Jan 01 2004
a(n) = A059304(n-1), n>1. [R. J. Mathar, Sep 29 2008]
a(n) ~ 2^(3*n-3)/sqrt(Pi*n). - Vaclav Kotesovec, Sep 28 2019
E.g.f.: x * (exp(4*x) * (BesselI(0,4*x) - BesselI(1,4*x)) - 1). - Ilya Gutkovskiy, Nov 03 2021
From Amiram Eldar, Jan 16 2024: (Start)
Sum_{n>=2} 1/a(n) = 1/7 + 8*arcsin(1/(2*sqrt(2)))/(7*sqrt(7)).
Sum_{n>=2} (-1)^n/a(n) = 1/9 + 4*log(2)/27. (End)
Showing 1-2 of 2 results.

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