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%I #38 Nov 07 2022 12:55:11
%S 1,0,1,0,1,1,0,0,2,1,0,0,2,3,1,0,0,0,5,4,1,0,0,0,5,9,5,1,0,0,0,0,14,
%T 14,6,1,0,0,0,0,14,28,20,7,1,0,0,0,0,0,42,48,27,8,1,0,0,0,0,0,42,90,
%U 75,35,9,1,0,0,0,0,0,0,132,165,110,44,10,1
%N Another version of Catalan triangle A009766.
%C Triangle T(n,k), 0 <= k <= n, read by rows, given by [0, 1, -1, 0, 0, 1, -1, 0, 0, 1, -1, 0, 0, ...] DELTA [1, 0, 0, -1, 1, 0, 0, -1, 1, 0, 0, -1, 1, ...] where DELTA is the operator defined in A084938.
%C Aerated version gives A165408. - _Philippe Deléham_, Sep 22 2009
%C T(n,k) is the number of length n left factors of Dyck paths having k up steps. Example: T(5,4)=4 because we have UDUUU, UUDUU, UUUDU, and UUUUD, where U=(1,1) and D=(1,-1). - _Emeric Deutsch_, Jun 19 2011
%C With zeros omitted: 1,1,1,1,2,1,2,3,1,5,4,1,... = A008313. - _Philippe Deléham_, Nov 02 2011
%H Alois P. Heinz, <a href="/A120730/b120730.txt">Rows n = 0..200, flattened</a>
%F G.f.: G(t,z) = 4*z/((2*z-1+sqrt(1-4*t*z^2))*(1+sqrt(1-4*t*z^2)). - _Emeric Deutsch_, Jun 19 2011
%F Sum_{k=0..n} x^k*T(n,n-k) = A001405(n), A126087(n), A128386(n), A121724(n), A128387(n), A132373(n), A132374(n), A132375(n), A121725(n) for x=1,2,3,4,5,6,7,8,9 respectively. [corrected by _Philippe Deléham_, Oct 16 2008]
%F T(2*n,n) = A000108(n); A000108: Catalan numbers.
%F From _Philippe Deléham_, Oct 18 2008: (Start)
%F Sum_{k=0..n} T(n,k)^2 = A000108(n) and Sum_{n>=k} T(n,k) = A000108(k+1).
%F Sum_{k=0..n} T(n,k)^3 = A003161(n).
%F Sum_{k=0..n} T(n,k)^4 = A129123(n). (End)
%F Sum_{k=0..n}, T(n,k)*x^k = A000007(n), A001405(n), A151281(n), A151162(n), A151254(n), A156195(n), A156361(n), A156362(n), A156566(n), A156577(n) for x=0,1,2,3,4,5,6,7,8,9 respectively. - _Philippe Deléham_, Feb 10 2009
%F From _G. C. Greubel_, Nov 07 2022: (Start)
%F T(n, k) = 0 if n > 2*k, otherwise binomial(n, k)*(2*k-n+1)/(k+1).
%F Sum_{k=0..n} (-1)^k*T(n,k) = A105523(n).
%F Sum_{k=0..n} (-1)^k*T(n,k)^2 = -A132889(n), n >= 1.
%F Sum_{k=0..floor(n/2)} T(n-k, k) = A357654(n).
%F T(n, n-1) = A001477(n).
%F T(n, n-2) = [n=2] + A000096(n-3), n >= 2.
%F T(n, n-3) = 2*[n<5] + A005586(n-5), n >= 3.
%F T(n, n-4) = 5*[n<7] - 2*[n=4] + A005587(n-7), n >= 4.
%F T(2*n+1, n+1) = A000108(n+1), n >= 0.
%F T(2*n-1, n+1) = A099376(n-1), n >= 1. (End)
%e As a triangle, this begins:
%e 1;
%e 0, 1;
%e 0, 1, 1;
%e 0, 0, 2, 1;
%e 0, 0, 2, 3, 1;
%e 0, 0, 0, 5, 4, 1;
%e 0, 0, 0, 5, 9, 5, 1;
%e 0, 0, 0, 0, 14, 14, 6, 1;
%e ...
%p G := 4*z/((2*z-1+sqrt(1-4*z^2*t))*(1+sqrt(1-4*z^2*t))): Gser := simplify(series(G, z = 0, 13)): for n from 0 to 12 do P[n] := sort(coeff(Gser, z, n)) end do: for n from 0 to 12 do seq(coeff(P[n], t, k), k = 0 .. n) end do; # yields sequence in triangular form # _Emeric Deutsch_, Jun 19 2011
%p # second Maple program:
%p b:= proc(x, y) option remember; `if`(y<0 or y>x, 0,
%p `if`(x=0, 1, add(b(x-1, y+j), j=[-1, 1])))
%p end:
%p T:= (n, k)-> b(n, 2*k-n):
%p seq(seq(T(n, k), k=0..n), n=0..14); # _Alois P. Heinz_, Oct 13 2022
%t b[x_, y_]:= b[x, y]= If[y<0 || y>x, 0, If[x==0, 1, Sum[b[x-1, y+j], {j, {-1, 1}}] ]];
%t T[n_, k_] := b[n, 2 k - n];
%t Table[Table[T[n, k], {k, 0, n}], {n, 0, 14}] // Flatten (* _Jean-François Alcover_, Oct 21 2022, after _Alois P. Heinz_ *)
%t T[n_, k_]:= If[n>2*k, 0, Binomial[n, k]*(2*k-n+1)/(k+1)];
%t Table[T[n, k], {n,0,13}, {k,0,n}]//Flatten (* _G. C. Greubel_, Nov 07 2022 *)
%o (Magma)
%o A120730:= func< n,k | n gt 2*k select 0 else Binomial(n, k)*(2*k-n+1)/(k+1) >;
%o [A120730(n,k): k in [0..n], n in [0..13]]; // _G. C. Greubel_, Nov 07 2022
%o (SageMath)
%o def A120730(n,k): return 0 if (n>2*k) else binomial(n, k)*(2*k-n+1)/(k+1)
%o flatten([[A120730(n,k) for k in range(n+1)] for n in range(14)]) # _G. C. Greubel_, Nov 07 2022
%Y Cf. A000007, A000096, A000108, A001405, A001477, A003161, A005586, A005587.
%Y Cf. A008313, A009766, A039598, A039599, A084938, A099376, A105523, A121725.
%Y Cf. A126087, A128386, A121724, A128387, A129123, A132373, A132374, A132375.
%Y Cf. A132889, A151162, A151254, A151281, A156195, A156361, A156362, A156566.
%Y Cf. A156577, A165408, A357654.
%K nonn,tabl
%O 0,9
%A _Philippe Deléham_, Aug 17 2006, corrected Sep 15 2006
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