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Riordan array (1, x*f(x)) where f(x)is the g.f. of A006318.
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%I #35 Oct 28 2024 04:52:22

%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 41586,33028,17718,7432,2490,652,126,16,1,0,206098,164512,89898,39152,14002,4080,938,160,18,1

%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. C. Greubel, <a href="/A122538/b122538.txt">Rows n = 0..100 of the triangle, flattened</a>

%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,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 Sum_{k=0..n} T(n, k) = A001003(n).

%F From _G. C. Greubel_, Oct 27 2024: (Start)

%F T(2*n, n) = A103885(n).

%F Sum_{k=0..n} (-1)^k*T(n, k) = -A001003(n-1).

%F Sum_{k=0..floor(n/2)} T(n-k, k) = [n=0] + 0*[n=1] + A006603(n-2)*[n>1]. (End)

%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 0, 394, 304, 146, 48, 10, 1;

%e 0, 1806, 1412, 714, 264, 70, 12, 1;

%e 0, 8558, 6752, 3534, 1408, 430, 96, 14, 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,0,12}, {k,0,n}]//Flatten (* _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..12): print(A122538_row(n)) # _Peter Luschny_, Mar 16 2016

%o (Magma)

%o function T(n,k) // T = A122538

%o if k eq 0 then return 0^n;

%o elif k eq n then return 1;

%o else return T(n-1,k-1) + T(n-1,k) + T(n,k+1);

%o end if;

%o end function;

%o [T(n,k): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Oct 27 2024

%Y Another version : A080247, A080245, A033877.

%Y Columns: A000007, A006318, A006319, A006320, A006321.

%Y Diagonals: A000012, A005843, A054000.

%Y Sums include: A001003 (row and alternating sign), A006603 (diagonal).

%Y Cf. A103885.

%K nonn,tabl

%O 0,5

%A _Philippe Deléham_, Sep 18 2006