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A180955
Triangle read by rows T(n,k) = numerators of A180955/A180956.
2
1, 1, 1, 3, 1, 1, 5, 3, 1, 1, 35, 5, 3, 1, 1, 63, 35, 5, 3, 1, 1, 231, 63, 35, 5, 3, 1, 1, 429, 231, 63, 35, 5, 3, 1, 1, 6435, 429, 231, 63, 35, 5, 3, 1, 1, 12155, 6435, 429, 231, 63, 35, 5, 3, 1, 1, 46189, 12155, 6435, 429, 231, 63, 35, 5, 3, 1, 1, 88179, 46189, 12155, 6435, 429, 231, 63, 35, 5, 3, 1, 1
OFFSET
0,4
COMMENTS
Consider the fractional triangle A180955/A180956, call it triangle A. Consider also a triangle defined by k=1: T(n,1)=1, k>1 and n>=k: T(n,k)= any random number, else 0, call it triangle B. Calculate the matrix inverse of triangle B, call it triangle C. Multiply C with A, call the result triangle D. Calculate the matrix inverse of D, call it triangle E. Then the first column in both matrix A and matrix E will have the fraction A001790/A046161 in the first column.
FORMULA
From G. C. Greubel, Sep 22 2024: (Start)
T(n, k) = A001790(n-k) = numerator(binomial(2*(n-k), n-k)/4^(n-k)).
T(n, 0) = T(2*n, n) = A001790(n).
Sum_{k=0..n} (-1)^k*T(n, k) = A001790(n+1) + Sum_{j=0..n+1} (-1)^(n+j)*A001790(j).
Sum_{k=0..floor(n/2)} T(n-k, k) = (1/2)*Sum_{j=0..n} (1+(-1)^(n+j))*A001790(j). (End)
EXAMPLE
Triangle starts:
1;
1, 1;
3, 1, 1;
5, 3, 1, 1;
35, 5, 3, 1, 1;
63, 35, 5, 3, 1, 1;
231, 63, 35, 5, 3, 1, 1;
429, 231, 63, 35, 5, 3, 1, 1;
6435, 429, 231, 63, 35, 5, 3, 1, 1;
12155, 6435, 429, 231, 63, 35, 5, 3, 1, 1;
46189, 12155, 6435, 429, 231, 63, 35, 5, 3, 1, 1;
88179, 46189, 12155, 6435, 429, 231, 63, 35, 5, 3, 1, 1;
MATHEMATICA
A180955[n_, k_]:= Numerator[Binomial[2*(n-k), n-k]/4^(n-k)];
Table[A180955[n, k], {n, 0, 12}, {k, 0, n}]//Flatten (* G. C. Greubel, Sep 22 2024 *)
PROG
(Magma)
A180955:= func< n, k | Numerator((n-k+1)*Catalan(n-k)/4^(n-k)) >;
[A180955(n, k): k in [0..n], n in [0..12]]; // G. C. Greubel, Sep 22 2024
(SageMath)
def A180955(n, k): return numerator(binomial(2*(n-k), n-k)/4^(n-k))
flatten([[A180955(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 22 2024
STATUS
approved