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A122538 Riordan array (1, x*f(x)) where f(x)is the g.f. of A006318.

%I A122538 #35 Oct 28 2024 04:52:22
%S A122538 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 A122538 0,1806,1412,714,264,70,12,1,0,8558,6752,3534,1408,430,96,14,1,0,
%U A122538 41586,33028,17718,7432,2490,652,126,16,1,0,206098,164512,89898,39152,14002,4080,938,160,18,1
%N A122538 Riordan array (1, x*f(x)) where f(x)is the g.f. of A006318.
%C A122538 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 A122538 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 A122538 G. C. Greubel, <a href="/A122538/b122538.txt">Rows n = 0..100 of the triangle, flattened</a>
%H A122538 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 A122538 T(n,k) = T(n-1,k-1) + T(n-1,k) + T(n,k+1) if k > 0, with T(n, 0) = 0^n, and T(n, n) = 1.
%F A122538 Sum_{k=0..n} T(n, k) = A001003(n).
%F A122538 From _G. C. Greubel_, Oct 27 2024: (Start)
%F A122538 T(2*n, n) = A103885(n).
%F A122538 Sum_{k=0..n} (-1)^k*T(n, k) = -A001003(n-1).
%F A122538 Sum_{k=0..floor(n/2)} T(n-k, k) = [n=0] + 0*[n=1] + A006603(n-2)*[n>1]. (End)
%e A122538 Triangle begins:
%e A122538   1;
%e A122538   0,    1:
%e A122538   0,    2,    1;
%e A122538   0,    6,    4,    1;
%e A122538   0,   22,   16,    6,    1;
%e A122538   0,   90,   68,   30,    8,   1;
%e A122538   0,  394,  304,  146,   48,  10,  1;
%e A122538   0, 1806, 1412,  714,  264,  70, 12,  1;
%e A122538   0, 8558, 6752, 3534, 1408, 430, 96, 14, 1;
%e A122538 Production matrix is:
%e A122538   0...1
%e A122538   0...2...1
%e A122538   0...2...2...1
%e A122538   0...2...2...2...1
%e A122538   0...2...2...2...2...1
%e A122538   0...2...2...2...2...2...1
%e A122538   0...2...2...2...2...2...2...1
%e A122538   0...2...2...2...2...2...2...2...1
%e A122538   0...2...2...2...2...2...2...2...2...1
%e A122538   ... - _Philippe Deléham_, Feb 09 2014
%t A122538 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 A122538 Table[T[n,k], {n,0,12}, {k,0,n}]//Flatten (* _Jean-François Alcover_, Jun 13 2019 *)
%o A122538 (Sage)
%o A122538 def A122538_row(n):
%o A122538     @cached_function
%o A122538     def prec(n, k):
%o A122538         if k==n: return 1
%o A122538         if k==0: return 0
%o A122538         return prec(n-1,k-1)-2*sum(prec(n,k+i-1) for i in (2..n-k+1))
%o A122538     return [(-1)^(n-k)*prec(n, k) for k in (0..n)]
%o A122538 for n in (0..12): print(A122538_row(n)) # _Peter Luschny_, Mar 16 2016
%o A122538 (Magma)
%o A122538 function T(n,k) // T = A122538
%o A122538   if k eq 0 then return 0^n;
%o A122538   elif k eq n then return 1;
%o A122538   else return T(n-1,k-1) + T(n-1,k) + T(n,k+1);
%o A122538   end if;
%o A122538 end function;
%o A122538 [T(n,k): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Oct 27 2024
%Y A122538 Another version : A080247, A080245, A033877.
%Y A122538 Columns: A000007, A006318, A006319, A006320, A006321.
%Y A122538 Diagonals: A000012, A005843, A054000.
%Y A122538 Sums include: A001003 (row and alternating sign), A006603 (diagonal).
%Y A122538 Cf. A103885.
%K A122538 nonn,tabl
%O A122538 0,5
%A A122538 _Philippe Deléham_, Sep 18 2006