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A093730
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Antidiagonal sums of triangle A093729, which enumerates the number of nodes in the tree of tournament sequences.
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2
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1, 1, 2, 5, 18, 102, 949, 14731, 386060, 17323052, 1351157580, 185867701560, 45682244004244, 20283964291276804, 16423005586691362832, 24434416299840231799694, 67236458264587977465709983
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OFFSET
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0,3
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COMMENTS
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Related to A008934 (the number of tournament sequences).
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LINKS
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FORMULA
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a(n) = Sum_{k=0..n} A093729(n-k, k).
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MATHEMATICA
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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 *)
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PROG
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(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
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))
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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