0,3

Related to A008934 (the number of tournament sequences).

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.

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

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 k<n+1: return T(n, k-1) - T(n-1, k) + T(n-1, 2*k-1) + T(n-1, 2*k)

else: return sum((-1)^(j-1)*binomial(n+1, j)*T(n, k-j) for j in range(1, n+2))

def A093730(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

Cf. A008934, A093729.

Sequence in context: A007127 A279207 A005639 * A366449 A304918 A007769

Adjacent sequences: A093727 A093728 A093729 * A093731 A093732 A093733

Paul D. Hanna, Apr 14 2004

approved