A157243 - OEIS

%I #12 Oct 05 2024 20:00:43

%S 1,1,1,1,3,1,1,6,6,1,1,10,336,10,1,1,15,825,825,15,1,1,21,1716,197676,

%T 1716,21,1,1,28,3185,512050,512050,3185,28,1,1,36,5440,1163800,

%U 294296640,1163800,5440,36,1,1,45,8721,2395575,778076145,778076145,2395575,8721,45,1

%N Triangle T(n, k) = A001263(n*f(n,k) + 1, f(n,k) + 1), where f(n, k) = k if k <= floor(n/2) otherwise n-k, read by rows.

%H G. C. Greubel, <a href="/A157243/b157243.txt">Rows n = 0..50 of the triangle, flattened</a>

%F T(n, k) = A001263(n*f(n,k) + 1, f(n,k) + 1), where f(n, k) = k if k <= floor(n/2) otherwise n-k.

%F T(n, n-k) = T(n, k).

%F T(n, 1) = A000217(n). - _G. C. Greubel_, Jan 11 2022

%e Triangle begins as:

%e   1;

%e   1,  1;

%e   1,  3,    1;

%e   1,  6,    6,       1;

%e   1, 10,  336,      10,         1;

%e   1, 15,  825,     825,        15,         1;

%e   1, 21, 1716,  197676,      1716,        21,       1;

%e   1, 28, 3185,  512050,    512050,      3185,      28,    1;

%e   1, 36, 5440, 1163800, 294296640,   1163800,    5440,   36,  1;

%e   1, 45, 8721, 2395575, 778076145, 778076145, 2395575, 8721, 45, 1;

%t f[n_, k_]:= If[k<=Floor[n/2], k, n-k];

%t A001263[n_, k_]:= Binomial[n-1,k-1]*Binomial[n,k]/(n-k+1);

%t T[n_, k_]:= A001263[n*f[n,k] +1, f[n,k] +1];

%t Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten (* modified by _G. C. Greubel_, Jan 11 2022 *)

%o (Magma)

%o f:= func< n,k | k le Floor(n/2) select k else n-k >;

%o A001263:= func< n,k | Binomial(n-1,k-1)*Binomial(n,k)/(n-k+1) >;

%o [A001263(n*f(n,k)+1, f(n,k)+1): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Jan 11 2022

%o (Sage)

%o def f(n,k): return k if (k <= (n//2)) else n-k

%o def A001263(n,k): return binomial(n-1,k-1)*binomial(n,k)/(n-k+1)

%o flatten([[A001263(n*f(n,k)+1, f(n,k)+1) for k in (0..n)] for n in (0..12)]) # _G. C. Greubel_, Jan 11 2022

%Y Cf. A000217, A001263, A157219, A157221.

%K nonn,tabl,changed

%O 0,5

%A _Roger L. Bagula_, Feb 25 2009

%E Edited by _G. C. Greubel_, Jan 11 2022