A180956 - OEIS

%I #8 Sep 24 2024 09:29:20

%S 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,

%T 2,1,2048,1024,256,128,16,8,2,1,32768,2048,1024,256,128,16,8,2,1,

%U 65536,32768,2048,1024,256,128,16,8,2,1,262144,65536,32768,2048,1024,256,128,16,8,2,1

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

%H G. C. Greubel, <a href="/A180956/b180956.txt">Rows n = 0..50 of the triangle, flattened</a>

%F From _G. C. Greubel_, Sep 23 2024: (Start)

%F T(n, k) = A046161(n-k) = denominator(binomial(2*(n-k), n-k)/4^(n-k)).

%F T(n, 0) = T(2*n, n) = A046161(n).

%F Sum_{k=0..n} T(n, k) = Sum_{j=0..n} A046161(j).

%F Sum_{k=0..n} (-1)^k*T(n, k) = A046161(n+1) + Sum_{j=0..n+1} (-1)^(n+j)*A046161(j).

%F 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)

%e Triangle starts:

%e        1;

%e        2,     1;

%e        8,     2,     1;

%e       16,     8,     2,    1;

%e      128,    16,     8,    2,    1;

%e      256,   128,    16,    8,    2,   1;

%e     1024,   256,   128,   16,    8,   2,   1;

%e     2048,  1024,   256,  128,   16,   8,   2,  1;

%e    32768,  2048,  1024,  256,  128,  16,   8,  2,  1;

%e    65536, 32768,  2048, 1024,  256, 128,  16,  8,  2,  1;

%e   262144, 65536, 32768, 2048, 1024, 256, 128, 16,  8,  2,  1;

%t A180956[n_, k_]:= Denominator[Binomial[2*(n-k), n-k]/4^(n-k)];

%t Table[A180956[n,k], {n,0,12}, {k,0,n}]//Flatten (* _G. C. Greubel_, Sep 23 2024 *)

%o (Magma)

%o A180956:= func< n,k | Denominator((n-k+1)*Catalan(n-k)/4^(n-k)) >;

%o [A180956(n,k): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Sep 22 2024

%o (SageMath)

%o def A180956(n,k): return denominator(binomial(2*(n-k), n-k)/4^(n-k))

%o flatten([[A180956(n,k) for k in range(n+1)] for n in range(13)]) # _G. C. Greubel_, Sep 22 2024

%Y Cf. A001790, A046161, A180955.

%K nonn,tabl

%O 0,2

%A _Mats Granvik_, Sep 28 2010

%E Offset changed by _G. C. Greubel_, Sep 23 2024