A108080 a(n) = Sum_{i=0..n} binomial(2*n+i,n-i).

1, 3, 12, 50, 211, 895, 3805, 16193, 68940, 293526, 1249622, 5318976, 22634700, 96296410, 409573584, 1741574006, 7403616923, 31466106703, 133704121665, 568008916093, 2412570019447, 10245302874071, 43500597657111, 184670002546295, 783850164628721, 3326671128027805, 14116630429874265

Offset

0,2

Comments

Apparently a bisection of A026847.

Row sums of A159965. - Paul Barry, Apr 28 2009

The observation above is true (see Fried link). - Sela Fried, Mar 13 2026

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Sela Fried, Proof of a conjecture stated in A108080, 2026.

Formula

From Paul Barry, Apr 28 2009: (Start)

G.f.: x/(x*sqrt(1-4x)-(1-2x-(1-3x)*c(x))), c(x) the g.f. of A000108.

a(n) = Sum_{k=0..n} Sum_{j=0..n} C(n+k,j-k)*C(n,j). (End)

From Paul Barry, Sep 07 2009: (Start)

G.f.: (1/sqrt(1-4x))*(1/(1-xc(x)^3)), c(x) the g.f. of A000108.

a(n) = Sum_{k=0..n} C(2n,n-k)*F(k+1) = Sum_{k=0..n} C(2n,k)*F(n-k+1).

a(n) = Sum_{k=0..n} C(2k,k) * A165201(n-k). (End)

From Vaclav Kotesovec, Oct 24 2012: (Start)

Recurrence: n*(17*n-93)*a(n) = 4*(34*n^2 - 189*n + 98)*a(n-1) - 5*(51*n^2 - 271*n + 252)*a(n-2) - 4*(17*n^2 - 184*n + 406)*a(n-3) + 44*(2*n-7) * a(n-4).

a(n) ~ 1/2*(1+1/sqrt(5))*(sqrt(5)+2)^n. (End)

a(n) = binomial(2*n, n)*hypergeom([1, -n, 1+2*n], [(1+n)/2, 1+n/2], -1/4). - Stefano Spezia, Jun 17 2025

Mathematica

CoefficientList[Series[x/(x*Sqrt[1-4*x]-(1-2*x-(1-3*x)*(1-Sqrt[1-4*x])/(2*x))), {x, 0, 20}], x] (* Vaclav Kotesovec, Oct 24 2012 *)

Prog

(PARI) x='x+O('x^66); Vec(x/(x*sqrt(1-4*x)-(1-2*x-(1-3*x)*(1-sqrt(1-4*x))/(2*x)))) \\ Joerg Arndt, May 15 2013

Crossrefs

Cf. A000108, A026847, A159965, A165201.

Sequence in context: A229665 A092443 A356280 * A113441 A119976 A074547

Adjacent sequences: A108077 A108078 A108079 * A108081 A108082 A108083

Keyword

nonn

Author

Ralf Stephan, Jun 03 2005

Status

approved