A118928 a(n) = Sum_{k=0..floor(n/2)} C(n-k,k)*C(n-k,k+1)/(n-k) * a(k), with a(0)=1.

1, 1, 1, 2, 4, 8, 17, 38, 92, 238, 643, 1790, 5076, 14573, 42241, 123484, 364052, 1082602, 3247759, 9829820, 30019326, 92517644, 287805801, 903822922, 2865339252, 9168572009, 29601077285, 96377791839, 316264456921

Offset: 0

Author

Paul D. Hanna, May 06 2006

Keywords

• nonn

Comments

Invariant column vector V under matrix product A089732 *V = V: a(n) = Sum_{k=0,[n/2]} A089732 (n,k)*a(k), where A089732(n,k) = number of peakless Motzkin paths of length n having k (1,1) steps.

Links

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

Crossrefs

Cf. A089732.

Programs

Mathematica

a[n_]:= a[n]= If[n==0, 1, Sum[Binomial[n-k, k]*Binomial[n-k, k+1]*a[k]/(n-k), {k, 0, Floor[n/2]}]];
Table[a[n], {n, 0, 30}] (* G. C. Greubel, Nov 24 2021 *)

PARI

{a(n)=if(n==0,1,sum(k=0,n\2,binomial(n-k,k)*binomial(n-k,k+1)/(n-k)*a(k)))}

Sage

@CachedFunction
def A118928(n):
    if (n==0): return 1
    else: return sum( binomial(n-k, k)*binomial(n-k, k+1)*A118928(k)/(n-k) for k in (0..n//2) )
[A118928(n) for n in (0..30)] # G. C. Greubel, Nov 24 2021

Formula

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