A029609 Central numbers in the (2,3)-Pascal triangle A029600.

1, 5, 15, 50, 175, 630, 2310, 8580, 32175, 121550, 461890, 1763580, 6760390, 26001500, 100291500, 387793800, 1502700975, 5834015550, 22687838250, 88363159500, 344616322050, 1345644686100, 5260247409300, 20583576819000, 80619009207750, 316026516094380, 1239796332370260

Offset: 0

Author

Mohammad K. Azarian

Keywords

• nonn
• easy

Comments

For n > 0 also central terms of (1,4)-Pascal triangle A095666. - Reinhard Zumkeller, Apr 08 2012

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 0..1000

Crossrefs

Cf. A000108, A029600, A095666.

Programs

Haskell

a029609 n = a029600 (2*n) n  -- Reinhard Zumkeller, Apr 08 2012

Mathematica

a[n_]=(5*Binomial[2*n,n]-3KroneckerDelta[n,0])/2; Array[a,27,0] (* Stefano Spezia, Feb 14 2025 *)

Formula

From Peter Bala, Aug 16 2011: (Start)

a(n) = (5/2)*binomial(2*n,n) for n >= 1.

O.g.f.: -3/2+5/2*1/sqrt(1-4*x) = 1+5*x+15*x^2+50*x^3+... = 1+5*x*d/dx(log(C(x))), where C(x) is the o.g.f. for the Catalan numbers A000108. (End)

E.g.f.: (5*exp(2*x)*BesselI(0, 2*x) - 3)/2. - Stefano Spezia, Feb 14 2025

Extensions

More terms from James Sellers