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%I #15 Aug 25 2023 08:28:20
%S 1,0,1,0,0,1,0,1,0,1,0,0,2,0,1,0,3,0,3,0,1,0,0,7,0,4,0,1,0,12,0,12,0,
%T 5,0,1,0,0,30,0,18,0,6,0,1,0,55,0,55,0,25,0,7,0,1,0,0,143,0,88,0,33,0,
%U 8,0,1
%N Riordan array (1, 2*sqrt(3)*sin(arcsin(3*sqrt(3)*x/2)/3)/3).
%H G. C. Greubel, <a href="/A124305/b124305.txt">Rows n = 0..50 of the triangle, flattened</a>
%F Sum_{k=0..n} T(n, k) = A047749(n).
%F Sum_{k=0..floor(n/2)} T(n-k, k) = (1/2)*(1 + (-1)^n)*A098746(n/2).
%F From _G. C. Greubel_, Aug 19 2023: (Start)
%F T(n, k) = (1/2)*(1 + (-1)^(n-k))*(k/n)*binomial(n + (n-k)/2 - 1, (n-k)/2), with T(0, 0) = 1.
%F T(n, n) = 1.
%F T(n, n-2) = A001477(n-2).
%F T(n, n-4) = A055998(n-4).
%F T(n, n-6) = A111396(n-6).
%F T(n, 0) = 0^n.
%F T(n, 1) = ((1-(-1)^n)/2)*A001764(floor((n-1)/2)).
%F T(n, 2) = ((1+(-1)^n)/2)*A006013(floor((n-2)/2)).
%F Sum_{k=0..n} (-1)^k*T(n, k) = (-1)^n * A047749(n). (End)
%e Triangle begins
%e 1,
%e 0, 1,
%e 0, 0, 1,
%e 0, 1, 0, 1,
%e 0, 0, 2, 0, 1,
%e 0, 3, 0, 3, 0, 1,
%e 0, 0, 7, 0, 4, 0, 1,
%e 0, 12, 0, 12, 0, 5, 0, 1
%e From _Paul Barry_, Sep 28 2009: (Start)
%e Production matrix is
%e 0, 1,
%e 0, 0, 1,
%e 0, 1, 0, 1,
%e 0, 0, 1, 0, 1,
%e 0, 1, 0, 1, 0, 1,
%e 0, 0, 1, 0, 1, 0, 1,
%e 0, 1, 0, 1, 0, 1, 0, 1,
%e 0, 0, 1, 0, 1, 0, 1, 0, 1,
%e 0, 1, 0, 1, 0, 1, 0, 1, 0, 1,
%e 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1 (End)
%t A124305[n_, k_]:= If[n==0, 1, (1/2)*(1+(-1)^(n-k))*(k/n)*Binomial[n +(n-k)/2 -1, (n-k)/2]];
%t Table[A124305[n, k], {n,0,12}, {k,0,n}]//Flatten (* _G. C. Greubel_, Aug 19 2023 *)
%o (Magma)
%o A124305:= func< n,k | n eq 0 select 1 else (1/2)*(1+(-1)^(n-k))*(k/n)*Binomial(n + Floor((n-k)/2) -1, n-1) >;
%o [A124305(n,k): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Aug 25 2023
%o (SageMath)
%o def A124305(n,k): return 1 if n==0 else ((n-k+1)%2)*k*binomial(n + (n-k)//2 -1, n-1)//n
%o flatten([[A124305(n,k) for k in range(n+1)] for n in range(13)]) # _G. C. Greubel_, Aug 25 2023
%Y Cf. A001477, A001764, A006013, A055998.
%Y Cf. A047749 (row sums), A098746 (diagonal sums), A124304 (inverse).
%K easy,nonn,tabl
%O 0,13
%A _Paul Barry_, Oct 25 2006
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