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.

A018224 a(n) = binomial(n, floor(n/2))^2 = A001405(n)^2.

Original entry on oeis.org

1, 1, 4, 9, 36, 100, 400, 1225, 4900, 15876, 63504, 213444, 853776, 2944656, 11778624, 41409225, 165636900, 590976100, 2363904400, 8533694884, 34134779536, 124408576656, 497634306624, 1828114918084, 7312459672336, 27043120090000, 108172480360000, 402335398890000
Offset: 0

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Author

N. J. A. Sloane

Keywords

Comments

a(n) is also the number of rooted two-vertex (or, dually, two-face) regular planar maps of valency n+1. - Valery A. Liskovets, Oct 19 2005
If A is a random matrix in USp(4) (4 X 4 complex matrices that are unitary and symplectic), then a(n)=(-1)^n*E[(tr(A^4))^n]. - Andrew V. Sutherland, Apr 01 2008
Number of square lattice walks with unit steps in all four directions (NSWE), starting at the origin, ending on the y-axis, and never going below the x-axis. Row sums of A378061. - Peter Luschny, Dec 08 2024

Examples

			The 9 lattice walks defined in the comments: 'NNN', 'NNS', 'NSN', 'NWE', 'NEW', 'WNE', 'WEN', 'ENW', 'EWN'.

Links

Crossrefs

Bisections are A002894 and A060150.
Cf. A000108, A001405, A056040, A113182, A138545, A378061.

Programs

  • Maple
    s := x -> (1+x)*EllipticK(x)/(x*Pi/2)-1/x:
seq(4^n*coeff(series(s(x),x,n+2),x,n),n=0..23); # Peter Luschny, Oct 14 2015
  • Mathematica
    (* Note that Mathematica uses a different definition of the EllipticK function. *)
CoefficientList[Series[(-Pi + (2 + 8 x) EllipticK[16 x^2])/(4 Pi x), {x,0,23}], x] (* Peter Luschny, Oct 14 2015 *)
Table[Binomial[n,Floor[n/2]]^2,{n,0,30}] (* Harvey P. Dale, Dec 02 2022 *)
  • PARI
    vector(50, n, n--; binomial(n, n\2)^2) \\ Altug Alkan, Oct 14 2015

Formula

E.g.f.: BesselI(0, 2*x)*(BesselI(0, 2*x)+BesselI(1, 2*x)). - Vladeta Jovovic, Jun 12 2005
G.f. (1+1/(4*x))*hypergeom([1/2, 1/2],[1],16*x^2)-1/(4*x). - Mark van Hoeij, Oct 13 2009
a(n) = (n!/(floor(n/2)!*floor((n+1)/2)!))^2. - Peter Luschny, Apr 29 2014
a(n) = A056040(n) * A056040(n+1) / (n+1). - Peter Luschny, Apr 29 2014
a(n) = 4^n*[x^n]((1+x)*EllipticK(x)/(x*Pi/2)-1/x). - Peter Luschny, Oct 14 2015
a(n) ~ 4^n*((2*n+3)/(2*n+1))^((-1)^n/2)/((n+1)*Pi/2). - Peter Luschny, Oct 14 2015
a(n) = Sum_{k=0..n} (-1)^k*binomial(n,k)*C(k)*binomial(2*n-2*k,n-k) where C(k) are Catalan numbers (A000108), see Prodinger. - Michel Marcus, Nov 19 2019
From Peter Bala, Jul 03 2023: (Start)
Right hand side of the binomial sum identity (1/2)*Sum_{k = 0..n+1} (-1)^k*4^(n+1-k)*binomial(n+1,k)*binomial(n+k,k)*binomial(2*k,k) = a(n).
a(n) = (1/2)*4^(n+1) * hypergeom([n+1, -n-1, 1/2], [1, 1], 1).
P-recursive:
(2*n - 1)*(n + 1)^2*a(n) = 4*(2*n^2 - 1)*a(n-1) + 16*(2*n + 1)*(n - 1)^2*a(n-2) with a(0) = a(1) = 1. (End)

A112944 Number of unrooted regular odd-valent planar maps with 2 vertices; maps are considered up to orientation-preserving homeomorphisms and the vertices are of valency 2n+1.

Original entry on oeis.org

1, 2, 7, 39, 308, 3013, 33300, 394340, 4878109, 62232321, 812825244, 10818489817, 146250545528, 2003199281223, 27747288947266, 388087900316025, 5474206895126243, 77795972452841542, 1112947041203866164, 16016508647052018408, 231727628211887783830, 3368855109532696440867
Offset: 0

Views

Author

Valery A. Liskovets, Oct 10 2005

Keywords

Examples

			There exist 2 planar maps with two 3-valent vertices: a map with three parallel edges and a map with one loop in each vertex and a link. Therefore a(1)=2.

Links

Crossrefs

Cf. A005470, A112945, A113181, A113182.

Programs

  • Mathematica
    a[n_] := (1/2) Binomial[2n, n] + (1/(4n+2)) Sum[EulerPhi[k] Binomial[2 Floor[n/k], Floor[n/k]]^2, {k, Divisors[2n+1]}];
Table[a[n], {n, 0, 21}] (* Jean-François Alcover, Jul 24 2018 *)
  • PARI
    a(n) = binomial(2*n, n)/2 + sumdiv(2*n+1, k, eulerphi(k)* binomial(2*(n\k), (n\k))^2)/(4*n+2); \\ Michel Marcus, Oct 14 2015

Formula

a(n) = (1/2)binomial(2n, n) + (1/(4n+2))sum_{k|(2n+1)}phi(k)* binomial(2*floor(n/k), floor(n/k))^2, where phi(k) is the Euler function A000010.

Extensions

More terms from Michel Marcus, Oct 14 2015

A113181 Number of unrooted two-vertex (or, dually, two-face) regular planar maps of even valency 2n considered up to orientation-preserving homeomorphism.

Original entry on oeis.org

1, 3, 14, 95, 859, 9130, 106039, 1297295, 16428300, 213388961, 2827645453, 38086408002, 520062618300, 7184570776213, 100256059855188, 1411319038583375, 20021022607979629, 285965560309310708, 4109498933510809561, 59380204746202961953, 862266486434574492404
Offset: 1

Views

Author

Valery A. Liskovets, Oct 19 2005

Keywords

Examples

			There exist 3 planar maps with two 4-valent vertices: a map with four parallel edges and two different maps with two parallel edges and one loop in each vertex. Therefore a(2)=3.

Links

Crossrefs

Cf. A000010, A113182, A112944.

Programs

  • Mathematica
    a[n_] := Binomial[2n, n]/4 + (1/(4n)) Sum[EulerPhi[k] Binomial[2n/k - 1, Floor[n/k]]^2, {k, Divisors[2n]}];
Array[a, 21] (* Jean-François Alcover, Jul 24 2018 *)
  • PARI
    a(n) = binomial(2*n, n)/4 + sumdiv(2*n, k, eulerphi(k)* binomial(2*n/k-1, (n\k))^2)/(4*n); \\ Michel Marcus, Oct 14 2015

Formula

a(n) = binomial(2*n,n)/4 + (1/(4*n))*Sum_{k|2*n} phi(k)*binomial((2*n/k)-1,floor(n/k))^2 where phi(k) is the Euler function A000010.

Extensions

More terms from Michel Marcus, Oct 14 2015
Showing 1-3 of 3 results.

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