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A124305 Riordan array (1, 2*sqrt(3)*sin(arcsin(3*sqrt(3)*x/2)/3)/3).

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