A124305 Riordan array (1, 2*sqrt(3)*sin(arcsin(3*sqrt(3)*x/2)/3)/3).
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, 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, 8, 0, 1
Offset: 0
Examples
Triangle begins 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, 5, 0, 1 From _Paul Barry_, Sep 28 2009: (Start) Production matrix is 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1 (End)
Links
- G. C. Greubel, Rows n = 0..50 of the triangle, flattened
Crossrefs
Programs
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Magma
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) >; [A124305(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Aug 25 2023
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Mathematica
A124305[n_, k_]:= If[n==0, 1, (1/2)*(1+(-1)^(n-k))*(k/n)*Binomial[n +(n-k)/2 -1, (n-k)/2]]; Table[A124305[n, k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Aug 19 2023 *)
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SageMath
def A124305(n,k): return 1 if n==0 else ((n-k+1)%2)*k*binomial(n + (n-k)//2 -1, n-1)//n flatten([[A124305(n,k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Aug 25 2023
Formula
Sum_{k=0..n} T(n, k) = A047749(n).
Sum_{k=0..floor(n/2)} T(n-k, k) = (1/2)*(1 + (-1)^n)*A098746(n/2).
From G. C. Greubel, Aug 19 2023: (Start)
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.
T(n, n) = 1.
T(n, n-2) = A001477(n-2).
T(n, n-4) = A055998(n-4).
T(n, n-6) = A111396(n-6).
T(n, 0) = 0^n.
T(n, 1) = ((1-(-1)^n)/2)*A001764(floor((n-1)/2)).
T(n, 2) = ((1+(-1)^n)/2)*A006013(floor((n-2)/2)).
Sum_{k=0..n} (-1)^k*T(n, k) = (-1)^n * A047749(n). (End)