A093730 Antidiagonal sums of triangle A093729, which enumerates the number of nodes in the tree of tournament sequences.

1, 1, 2, 5, 18, 102, 949, 14731, 386060, 17323052, 1351157580, 185867701560, 45682244004244, 20283964291276804, 16423005586691362832, 24434416299840231799694, 67236458264587977465709983

Offset: 0

Author

Paul D. Hanna, Apr 14 2004

Keywords

• nonn

Comments

Related to A008934 (the number of tournament sequences).

Links

• G. C. Greubel, Table of n, a(n) for n = 0..86
• M. Cook and M. Kleber, Tournament sequences and Meeussen sequences, Electronic J. Comb. 7 (2000), #R44.

Crossrefs

Cf. A008934, A093729.

Programs

Mathematica

T[n_, k_] := T[n, k] = If[n<0, 0, If[n==0, 1, If[k==0, 0, If[k<=n, T[n, k-1] - T[n-1, k] + T[n-1, 2*k-1] + T[n-1, 2*k], Sum[(-1)^(j-1) * Binomial[n+1, j]*T[n, k-j], {j, 1, n+1}]]]]]; a[n_] := Sum[T[n-k, k], {k, 0, n}]; Table[a[n], {n, 0, 16}] (* Jean-François Alcover, Oct 06 2016, translated from PARI *)

PARI

{T(n,k)=if(n<0,0,if(n==0,1,if(k==0,0, if(k<=n,T(n,k-1)-T(n-1,k)+T(n-1,2*k-1)+T(n-1,2*k), sum(j=1,n+1, (-1)^(j-1)*binomial(n+1,j)*T(n,k-j))))))}
a(n)=sum(k=0,n,T(n-k,k))

SageMath

@CachedFunction
def T(n, k): # T = A093729
    if n<0: return 0
    elif n==0: return 1
    elif k==0: return 0
    elif kA093730(n): return sum(T(n-k,k) for k in range(n+1))
[A093730(n) for n in range(31)] # G. C. Greubel, Feb 22 2024

Formula

a(n) = Sum_{k=0..n} A093729(n-k, k).