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A122538
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Riordan array (1, x*f(x)) where f(x)is the g.f. of A006318.
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1
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1, 0, 1, 0, 2, 1, 0, 6, 4, 1, 0, 22, 16, 6, 1, 0, 90, 68, 30, 8, 1, 0, 394, 304, 146, 48, 10, 1, 0, 1806, 1412, 714, 264, 70, 12, 1, 0, 8558, 6752, 3534, 1408, 430, 96, 14, 1, 0, 41586, 33028, 17718, 7432, 2490, 652, 126, 16, 1
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OFFSET
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0,5
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COMMENTS
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Triangle T(n,k), 0<=k<=n, read by rows, given by [0, 2, 1, 2, 1, 2, 1, ...] DELTA [1, 0, 0, 0, 0, 0, ...] where DELTA is the operator defined in A084938 . Inverse is Riordan array (1, x*(1-x)/(1+x)).
T(n, r) gives the number of [0,r]-covering hierarchies with n segments terminating at r (see Kreweras work). - Michel Marcus, Nov 22 2014
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LINKS
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G. Kreweras, Sur les hiérarchies de segments, Cahiers du Bureau Universitaire de Recherche Opérationnelle, Institut de Statistique, Université de Paris, #20 (1973), see page 15.
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FORMULA
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T(n,0) = 0^n, T(n,k) = T(n-1,k-1)+T(n-1,k)+T(n,k+1) if k>0.
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EXAMPLE
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Triangle begins:
1;
0, 1:
0, 2, 1;
0, 6, 4, 1;
0, 22, 16, 6, 1;
0, 90, 68, 30, 8, 1;
Production matrix is:
0...1
0...2...1
0...2...2...1
0...2...2...2...1
0...2...2...2...2...1
0...2...2...2...2...2...1
0...2...2...2...2...2...2...1
0...2...2...2...2...2...2...2...1
0...2...2...2...2...2...2...2...2...1
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MATHEMATICA
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T[n_, n_] = 1; T[_, 0] = 0; T[n_, k_] := T[n, k] = T[n-1, k-1] + T[n-1, k] + T[n, k+1];
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PROG
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(Sage)
@cached_function
def prec(n, k):
if k==n: return 1
if k==0: return 0
return prec(n-1, k-1)-2*sum(prec(n, k+i-1) for i in (2..n-k+1))
return [(-1)^(n-k)*prec(n, k) for k in (0..n)]
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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