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%I #29 Mar 07 2020 12:11:41
%S 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,
%T 0,1806,1412,714,264,70,12,1,0,8558,6752,3534,1408,430,96,14,1,0,
%U 41586,33028,17718,7432,2490,652,126,16,1
%N Riordan array (1, x*f(x)) where f(x)is the g.f. of A006318.
%C 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)).
%C 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
%H G. Kreweras, <a href="http://www.numdam.org/item?id=BURO_1973__20__3_0">Sur les hiérarchies de segments</a>, Cahiers du Bureau Universitaire de Recherche Opérationnelle, Institut de Statistique, Université de Paris, #20 (1973), see page 15.
%F T(n,0) = 0^n, T(n,k) = T(n-1,k-1)+T(n-1,k)+T(n,k+1) if k>0.
%e Triangle begins:
%e 1;
%e 0, 1:
%e 0, 2, 1;
%e 0, 6, 4, 1;
%e 0, 22, 16, 6, 1;
%e 0, 90, 68, 30, 8, 1;
%e Production matrix is:
%e 0...1
%e 0...2...1
%e 0...2...2...1
%e 0...2...2...2...1
%e 0...2...2...2...2...1
%e 0...2...2...2...2...2...1
%e 0...2...2...2...2...2...2...1
%e 0...2...2...2...2...2...2...2...1
%e 0...2...2...2...2...2...2...2...2...1
%e ... - _Philippe Deléham_, Feb 09 2014
%t 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];
%t Table[T[n, k], {n, 0, 9}, {k, 0, n}] (* _Jean-François Alcover_, Jun 13 2019 *)
%o (Sage)
%o def A122538_row(n):
%o @cached_function
%o def prec(n, k):
%o if k==n: return 1
%o if k==0: return 0
%o return prec(n-1,k-1)-2*sum(prec(n,k+i-1) for i in (2..n-k+1))
%o return [(-1)^(n-k)*prec(n, k) for k in (0..n)]
%o for n in (0..10): print(A122538_row(n)) # _Peter Luschny_, Mar 16 2016
%Y Another version : A080247, A080245, A033877.
%Y Columns: A000007, A006318, A006319, A006320, A006321.
%Y Diagonals: A000012, A005843, A054000.
%Y Cf. A001003 (row sums).
%K nonn,tabl
%O 0,5
%A _Philippe Deléham_, Sep 18 2006
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