A051631 -id:A051631 - cp's OEIS frontend

A024483 a(n) = binomial(2n, n) mod binomial(2n-2, n-1).

0, 2, 10, 42, 168, 660, 2574, 10010, 38896, 151164, 587860, 2288132, 8914800, 34767720, 135727830, 530365050, 2074316640, 8119857900, 31810737420, 124718287980, 489325340400, 1921133836440, 7547311500300, 29667795388452, 116686713634848

Offset: 2

Clark Kimberling

Apart from its root term -1: central terms of the triangle in A051631: a(n) = A051631(2*(n-1), n-1). - Reinhard Zumkeller, Nov 13 2011
Define an array m(i,j) by m(1,j)=m(j,1)=j*(j+1)/2 for j=0,1,2,3,... and m(i,j) = m(i,j-1) + m(i-1,j+1); the diagonal m(k,k) for k=1,2,3... gives the numbers in this sequence. - J. M. Bergot, May 02 2012
The central terms of triangle A051631 (including the root term -1) are given by (n-1)*(n+1)*Gamma(2*n+1)/Gamma(n+2)^2 with n >= 0. - Peter Luschny, Nov 24 2013
Index the sequence from n=0 so that a(0)=1, a(1)=0, a(2)=2, a(3)=10, ... a(n) is the number of walks using steps U=(1,1) and D=(1,-1) from the origin to (2n,0) that rise above and dip below the x axis. a(2) = 2 because we have: DUUD and UDDU. - Geoffrey Critzer, Jan 11 2014

Reinhard Zumkeller, Table of n, a(n) for n = 2..1000

Cf. A000108, A000984, A001622, A051631.

a024483 n = a051631 (2*(n-1)) (n-1) -- Reinhard Zumkeller, Nov 13 2011

seq((n-1)*binomial(2*n, n)/(n+1), n=1..25); # Zerinvary Lajos, Feb 28 2007

nn=20; d=(1-(1-4x)^(1/2))/(2x); Drop[CoefficientList[Series[1/(1-2x d)-2(d-1), {x,0,nn}],x],1] (* Geoffrey Critzer, Jan 11 2014 *)
Table[Mod[Binomial[2 n, n], Binomial[2 n - 2, n - 1]], {n, 2, 26}] (* Michael De Vlieger, Sep 13 2016 *)

def a(n): return n*(n-2)*factorial(2*(n-1))/factorial(n)^2
[a(n) for n in (2..26)]  # Peter Luschny, Nov 24 2013

a(n) = ((n-2)/n)*binomial(2*n-2, n-1) = (n-2)*A000108(n-1). - Vladeta Jovovic, Aug 03 2002
a(n) = 2*binomial(2n-3, n-3) = 2*A002054(n-2). - Ralf Stephan, Jan 15 2004
a(n) = A000984(n-1) - 2*A000108(n-1). - Geoffrey Critzer, Jan 11 2014
a(n) ~ 4^(n-1)/sqrt(Pi*n). - Ilya Gutkovskiy, Sep 13 2016
D-finite with recurrence n*a(n) +(-7*n+8)*a(n-1) +6*(2*n-5)*a(n-2)=0. - R. J. Mathar, Apr 27 2020
From Amiram Eldar, Mar 24 2022: (Start)
Sum_{n>=3} 1/a(n) = 5/6 - Pi/(9*sqrt(3)).
Sum_{n>=3} (-1)^(n+1)/a(n) = 26*sqrt(5)*log(phi)/25 - 7/10, where phi is the golden ratio (A001622). (End)

More terms from Zerinvary Lajos, Oct 02 2007

A116385 Expansion of e.g.f. Bessel_I(2,2x) + 2*Bessel_I(3,2x) + Bessel_I(4,2x).

Original entry on oeis.org

0, 0, 1, 2, 5, 10, 21, 42, 84, 168, 330, 660, 1287, 2574, 5005, 10010, 19448, 38896, 75582, 151164, 293930, 587860, 1144066, 2288132, 4457400, 8914800, 17383860, 34767720, 67863915, 135727830, 265182525, 530365050, 1037158320, 2074316640

Offset: 0

Paul Barry, Feb 12 2006

Third column of the Riordan array A116382.

Apart from its root term -1: central terms of the triangle in A051631: a(n) = A051631(n+1, [(n+1)/2]). - Reinhard Zumkeller, Nov 13 2011

Reinhard Zumkeller, Table of n, a(n) for n = 0..1000
Gennady Eremin, Dyck Numbers, II. Triplets and Rooted Trees in OEIS A036991, arXiv:2211.01135 [math.NT], 2022.

Cf. A001405.

a116385 n = a051631 (n+1) $ (n+1) `div` 2
-- Reinhard Zumkeller, Nov 13 2011

With[{nn=40},CoefficientList[Series[BesselI[2,2x]+2BesselI[3,2x]+ BesselI[ 4,2x],{x,0,nn}],x]Range[0,nn]!] (* Harvey P. Dale, Sep 14 2011 *)

a(n)= binomial(n+3, (n+3)\2) - 3*binomial(n+1, (n+1)\2) \\ Bill McEachen, Dec 12 2022

E.g.f.: (d/dx)(Bessel_I(3,2x),x) + 2*Bessel_I(3,2x).
a(n) = C(n+1,floor((n-2)/2))*(1+(-1)^n)/2 + C(n,floor((n-3)/2))*(1-(-1)^n).
Conjecture: (n+4)*a(n) -2*a(n-1) +(-7*n-8)*a(n-2) +6*a(n-3) +12*(n-2)*a(n-4)=0. - R. J. Mathar, Jun 13 2014
a(n) = A001405(n+3) - 3*A001405(n+1) (from Eremin link). - Bill McEachen, Dec 12 2022
G.f.: (-1 - x + x^2 + B(x) - 3*x^2*B(x))/x^3, where B(x) is the g.f. of A001405. - Gennady Eremin, Oct 09 2023

A276666 a(n) = (n-1)*Catalan(n).

Original entry on oeis.org

-1, 0, 2, 10, 42, 168, 660, 2574, 10010, 38896, 151164, 587860, 2288132, 8914800, 34767720, 135727830, 530365050, 2074316640, 8119857900, 31810737420, 124718287980, 489325340400, 1921133836440, 7547311500300, 29667795388452, 116686713634848, 459183826803800

Offset: 0

Peter Luschny, Sep 12 2016

sign

G. C. Greubel, Table of n, a(n) for n = 0..1000

A024483 is a variant of this sequence.

Cf. A000108, A000984, A001622, A051631, A056040, A057977, A232500.

Concatenation([-1], List([1..30], n-> 2*Binomial(2*n-1, n+1))); # G. C. Greubel, Aug 29 2019

[(n-1)*Catalan(n): n in [0..30]]; // Vincenzo Librandi, Sep 13 2016

f := (1-3*x)/(x*sqrt(1-4*x))-1/x:
series(f,x,29): seq(coeff(%,x,n), n=0..26);
A276666 := n -> (n^2-1)*(2*n)!/(n+1)!^2:
seq(A276666(n), n=0..26);

Table[(n - 1) CatalanNumber[n], {n, 0, 30}] (* Vincenzo Librandi, Sep 13 2016 *)

a(n) = if(n==0,-1, 2*binomial(2*n-1, n+1)); \\ G. C. Greubel, Aug 29 2019

A276666 = lambda n: (n - 1) * catalan_number(n)
[A276666(n) for n in range(27)]

a(n) = [x^n] (1-3*x)/(x*sqrt(1-4*x))-1/x.
a(n) = 4^n*(n-1)*hypergeom([3/2, -n], [2], 1).
a(n) = 4^n*(n-1)*JacobiP(n,1,-1/2-n,-1)/(n+1).
a(n) = (2*n)! [x^(2^n)]( BesselI(2,2*x) - (1+1/x)*BesselI(1,2*x) ).
a(n) = binomial(2*n,n) - 2*Catalan(n). (See Geoffrey Critzer's formula in A024483).
a(n) = A056040(2*n) - 2*A057977(2*n).
a(n) = A056040(2*n)*(1-2/(n+1)) = (n^2-1)*(2*n)!/(n+1)!^2.
a(n) = A232500(2*n).
a(n) = a(n-1)*2*(n-1)*(2*n-1)/((n-2)*(n+1)) for n > 2. - Chai Wah Wu, Sep 12 2016
a(n) = A024483(n+1) for n>0. - R. J. Mathar, Sep 13 2016
a(n) = A000984(n+1)-3*A000984(n). - Ezhilarasu Velayutham, Aug 27 2019
From Amiram Eldar, Mar 22 2022: (Start)
Sum_{n>=2} 1/a(n) = 5/6 - Pi/(9*sqrt(3)).
Sum_{n>=2} (-1)^n/a(n) = 26*sqrt(5)*log(phi)/25 - 7/10, where phi is the golden ratio (A001622). (End)

cp's OEIS Frontend

A024483 a(n) = binomial(2*n, n) mod binomial(2*n-2, n-1).

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A116385 Expansion of e.g.f. Bessel_I(2,2x) + 2*Bessel_I(3,2x) + Bessel_I(4,2x).

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A276666 a(n) = (n-1)*Catalan(n).

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A024483 a(n) = binomial(2n, n) mod binomial(2n-2, n-1).