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A180956
Triangle read by rows T(n,k) = denominators of A180955/A180956.
2
1, 2, 1, 8, 2, 1, 16, 8, 2, 1, 128, 16, 8, 2, 1, 256, 128, 16, 8, 2, 1, 1024, 256, 128, 16, 8, 2, 1, 2048, 1024, 256, 128, 16, 8, 2, 1, 32768, 2048, 1024, 256, 128, 16, 8, 2, 1, 65536, 32768, 2048, 1024, 256, 128, 16, 8, 2, 1, 262144, 65536, 32768, 2048, 1024, 256, 128, 16, 8, 2, 1
OFFSET
0,2
FORMULA
From G. C. Greubel, Sep 23 2024: (Start)
T(n, k) = A046161(n-k) = denominator(binomial(2*(n-k), n-k)/4^(n-k)).
T(n, 0) = T(2*n, n) = A046161(n).
Sum_{k=0..n} T(n, k) = Sum_{j=0..n} A046161(j).
Sum_{k=0..n} (-1)^k*T(n, k) = A046161(n+1) + Sum_{j=0..n+1} (-1)^(n+j)*A046161(j).
Sum_{k=0..floor(n/2)} T(n-k, k) = floor(n/2) + (1/2)*Sum_{j=0..n} (1+(-1)^(n+j)) * A046161(j). (End)
EXAMPLE
Triangle starts:
1;
2, 1;
8, 2, 1;
16, 8, 2, 1;
128, 16, 8, 2, 1;
256, 128, 16, 8, 2, 1;
1024, 256, 128, 16, 8, 2, 1;
2048, 1024, 256, 128, 16, 8, 2, 1;
32768, 2048, 1024, 256, 128, 16, 8, 2, 1;
65536, 32768, 2048, 1024, 256, 128, 16, 8, 2, 1;
262144, 65536, 32768, 2048, 1024, 256, 128, 16, 8, 2, 1;
MATHEMATICA
A180956[n_, k_]:= Denominator[Binomial[2*(n-k), n-k]/4^(n-k)];
Table[A180956[n, k], {n, 0, 12}, {k, 0, n}]//Flatten (* G. C. Greubel, Sep 23 2024 *)
PROG
(Magma)
A180956:= func< n, k | Denominator((n-k+1)*Catalan(n-k)/4^(n-k)) >;
[A180956(n, k): k in [0..n], n in [0..12]]; // G. C. Greubel, Sep 22 2024
(SageMath)
def A180956(n, k): return denominator(binomial(2*(n-k), n-k)/4^(n-k))
flatten([[A180956(n, k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Sep 22 2024
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Mats Granvik, Sep 28 2010
EXTENSIONS
Offset changed by G. C. Greubel, Sep 23 2024
STATUS
approved