A180956 Triangle read by rows T(n,k) = denominators of A180955/A180956.

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

Links

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

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

Cf. A001790, A046161, A180955.

Sequence in context: A205397 A020778 A118961 * A208921 A208660 A367024

Adjacent sequences: A180953 A180954 A180955 * A180957 A180958 A180959

Keyword

nonn,tabl

Author

Mats Granvik, Sep 28 2010

Extensions

Offset changed by G. C. Greubel, Sep 23 2024

Status

approved