A046224 -id:A046224 - cp's OEIS frontend

A185080 a(n) = 6 * binomial(2n,n-1) + binomial(2n-1,n).

7, 27, 100, 371, 1386, 5214, 19734, 75075, 286858, 1100138, 4232592, 16328942, 63146500, 244711260, 950094810, 3694876515, 14390571690, 56122547250, 219140635560, 856617714810, 3351878581740, 13127747882340, 51458942047500, 201869999056206, 792497263436676

Offset: 1

Reinhard Zumkeller, Dec 26 2012

nonn

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A000108, A007318, A024482, A046224, A046902.

a185080 n = 6 * a007318 (2 * n) (n - 1) + a007318 (2 * n - 1) n

[(13*n+1)*Catalan(n)/2: n in [1..40]]; // G. C. Greubel, Apr 03 2024

Table[6Binomial[2n,n-1]+Binomial[2n-1,n],{n,30}] (* Harvey P. Dale, Dec 28 2012 *)

[(13*n+1)*binomial(2*n,n)/(2*n+2) for n in range(1,41)] # G. C. Greubel, Apr 03 2024

a(n) = A046902(2*n,n) (Central terms of Clark's triangle).
a(n) = 6 * A007318(2*n,n-1) + A007318(2*n-1,n).
From G. C. Greubel, Apr 03 2024: (Start)
a(n) = (13*n+1)*A000108(n)/2.
a(n) = (2 + 22*n - 52*n^2)*a(n-1)/(12 - n - 13*n^2).
G.f.: ((6 - 11*x)*sqrt(1-4*x) - (1-4*x)*(6+x))/(2*x*(1-4*x)).
E.g.f.: (1/2)*(-1 + exp(2*x)*(BesselI(0, 2*x) + 12*BesselI(1, 2*x))).(End)

A261682 a(n) = 2^n+(1+(n mod 2)/2)*C(n+1,floor((n+(n mod 2))/2))-1.

Original entry on oeis.org

1, 4, 6, 16, 25, 61, 98, 232, 381, 889, 1485, 3433, 5811, 13339, 22818, 52072, 89845, 204001, 354521, 801421, 1401291, 3155299, 5546381, 12444841, 21977515, 49155331, 87167163, 194392627, 345994215, 769547191, 1374282018, 3049104232, 5461770405, 12090343921

Offset: 0

N. J. A. Sloane, Sep 04 2015

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Riccardo Biagioli, Frédéric Jouhet, and Philippe Nadeau, Combinatorics of fully commutative involutions in classical Coxeter groups, arXiv preprint arXiv:1411.4561 [math.CO] (2014). See Prop. 2.1.
Riccardo Biagioli, Frédéric Jouhet, and Philippe Nadeau, Combinatorics of fully commutative involutions in classical Coxeter groups, Discrete Math., 338 (2015), 2242-2259. See Prop. 2.1.

a:= n-> 2^n +((2+irem(n, 2))/2)*binomial(n+1, ceil(n/2))-1:
seq(a(n), n=0..40);  # Alois P. Heinz, Sep 05 2015

Table[If[EvenQ@ n, 2^n + Binomial[n + 1, n/2] - 1, 2^n + (3/2) Binomial[n + 1, (n + 1)/2] - 1], {n, 0, 33}] (* Michael De Vlieger, Sep 24 2015 *)

a(n) = if (n%2, 2^n+(3/2)*binomial(n+1,(n+1)/2)-1, 2^n+binomial(n+1,n/2)-1); \\ Michel Marcus, Sep 05 2015

If n mod 2 = 0, then a(n) = 2^n+binomial(n+1,n/2)-1, otherwise a(n) = 2^n+(3/2)*binomial(n+1,(n+1)/2)-1.
If n mod 2 = 0, then a(n+2) = a(n+1) + a(n) + A000108(n+1) - 2^n - 1; otherwise, a(n+2) = a(n+1) + a(n) + A046224(n+2) - 2^n - 1. - Eric Werley, Sep 16 2015
Conjecture: -(n+2)*(15*n^2-29*n-56)*a(n) +9*(5*n^3-3*n^2-24*n-20)*a(n-1) +2*(15*n^3-14*n^2-219*n+158)*a(n-2) +36*(-5*n^3+8*n^2+31*n-38)*a(n-3) +8*(n-2)*(15*n^2+n-70)*a(n-4)=0. - R. J. Mathar, Jan 04 2017

New name from Wesley Ivan Hurt, May 02 2021

cp's OEIS Frontend

A185080 a(n) = 6 * binomial(2n,n-1) + binomial(2n-1,n).

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A261682 a(n) = 2^n+(1+(n mod 2)/2)*C(n+1,floor((n+(n mod 2))/2))-1.

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cp's OEIS Frontend

A185080 a(n) = 6 * binomial(2*n,n-1) + binomial(2*n-1,n).

Views

Author

Keywords

Links

Crossrefs

Programs

Haskell

Magma

Mathematica

SageMath

Formula

A261682 a(n) = 2^n+(1+(n mod 2)/2)*C(n+1,floor((n+(n mod 2))/2))-1.

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Author

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Programs

Maple

Mathematica

PARI

Formula

Extensions

A185080 a(n) = 6 * binomial(2n,n-1) + binomial(2n-1,n).