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

%I A093730 #18 Feb 22 2024 09:05:14
%S A093730 1,1,2,5,18,102,949,14731,386060,17323052,1351157580,185867701560,
%T A093730 45682244004244,20283964291276804,16423005586691362832,
%U A093730 24434416299840231799694,67236458264587977465709983
%N A093730 Antidiagonal sums of triangle A093729, which enumerates the number of nodes in the tree of tournament sequences.
%C A093730 Related to A008934 (the number of tournament sequences).
%H A093730 G. C. Greubel, <a href="/A093730/b093730.txt">Table of n, a(n) for n = 0..86</a>
%H A093730 M. Cook and M. Kleber, <a href="https://doi.org/10.37236/1522">Tournament sequences and Meeussen sequences</a>, Electronic J. Comb. 7 (2000), #R44.
%F A093730 a(n) = Sum_{k=0..n} A093729(n-k, k).
%t A093730 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 *)
%o A093730 (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))))))}
%o A093730 a(n)=sum(k=0,n,T(n-k,k))
%o A093730 (SageMath)
%o A093730 @CachedFunction
%o A093730 def T(n, k): # T = A093729
%o A093730     if n<0: return 0
%o A093730     elif n==0: return 1
%o A093730     elif k==0: return 0
%o A093730     elif k<n+1: return T(n,k-1) - T(n-1,k) + T(n-1,2*k-1) + T(n-1,2*k)
%o A093730     else: return sum((-1)^(j-1)*binomial(n+1,j)*T(n, k-j) for j in range(1,n+2))
%o A093730 def A093730(n): return sum(T(n-k,k) for k in range(n+1))
%o A093730 [A093730(n) for n in range(31)] # _G. C. Greubel_, Feb 22 2024
%Y A093730 Cf. A008934, A093729.
%K A093730 nonn
%O A093730 0,3
%A A093730 _Paul D. Hanna_, Apr 14 2004