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%I #8 Sep 24 2024 03:09:37
%S 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,
%T 231,63,35,5,3,1,1,6435,429,231,63,35,5,3,1,1,12155,6435,429,231,63,
%U 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
%N Triangle read by rows T(n,k) = numerators of A180955/A180956.
%C 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.
%H G. C. Greubel, <a href="/A180955/b180955.txt">Rows n = 0..50 of the triangle, flattened</a>
%F From _G. C. Greubel_, Sep 22 2024: (Start)
%F T(n, k) = A001790(n-k) = numerator(binomial(2*(n-k), n-k)/4^(n-k)).
%F T(n, 0) = T(2*n, n) = A001790(n).
%F Sum_{k=0..n} (-1)^k*T(n, k) = A001790(n+1) + Sum_{j=0..n+1} (-1)^(n+j)*A001790(j).
%F Sum_{k=0..floor(n/2)} T(n-k, k) = (1/2)*Sum_{j=0..n} (1+(-1)^(n+j))*A001790(j). (End)
%e Triangle starts:
%e 1;
%e 1, 1;
%e 3, 1, 1;
%e 5, 3, 1, 1;
%e 35, 5, 3, 1, 1;
%e 63, 35, 5, 3, 1, 1;
%e 231, 63, 35, 5, 3, 1, 1;
%e 429, 231, 63, 35, 5, 3, 1, 1;
%e 6435, 429, 231, 63, 35, 5, 3, 1, 1;
%e 12155, 6435, 429, 231, 63, 35, 5, 3, 1, 1;
%e 46189, 12155, 6435, 429, 231, 63, 35, 5, 3, 1, 1;
%e 88179, 46189, 12155, 6435, 429, 231, 63, 35, 5, 3, 1, 1;
%t A180955[n_, k_]:= Numerator[Binomial[2*(n-k), n-k]/4^(n-k)];
%t Table[A180955[n,k], {n,0,12}, {k,0,n}]//Flatten (* _G. C. Greubel_, Sep 22 2024 *)
%o (Magma)
%o A180955:= func< n,k | Numerator((n-k+1)*Catalan(n-k)/4^(n-k)) >;
%o [A180955(n,k): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Sep 22 2024
%o (SageMath)
%o def A180955(n,k): return numerator(binomial(2*(n-k), n-k)/4^(n-k))
%o flatten([[A180955(n,k) for k in range(n+1)] for n in range(13)]) # _G. C. Greubel_, Sep 22 2024
%Y Cf. A001790, A046161, A180956.
%K nonn,tabl
%O 0,4
%A _Mats Granvik_, Sep 28 2010
%E Offset changed by _G. C. Greubel_, Sep 22 2024