A174346 - OEIS

%I #9 Sep 08 2022 08:45:51

%S 1,1,1,1,9,1,1,18,18,1,1,30,180,30,1,1,45,450,450,45,1,1,63,945,4725,

%T 945,63,1,1,84,1764,13230,13230,1764,84,1,1,108,3024,31752,142884,

%U 31752,3024,108,1,1,135,4860,68040,428652,428652,68040,4860,135,1

%N Triangle T(n, k) = (binomial(n-1, k-1)*binomial(n, k-1)/k) * ( 3^(k-1) if floor(n/2) >= k, otherwise 3^(n-k) ), read by rows.

%H G. C. Greubel, <a href="/A174346/b174346.txt">Rows n = 1..50 of the triangle, flattened</a>

%F T(n, k) = (binomial(n-1, k-1)*binomial(n, k-1)/k) * ( 3^(k-1) if floor(n/2) >= k, otherwise 3^(n-k) ).

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

%e Triangle begins as:

%e   1;

%e   1,   1;

%e   1,   9,    1;

%e   1,  18,   18,     1;

%e   1,  30,  180,    30,      1;

%e   1,  45,  450,   450,     45,      1;

%e   1,  63,  945,  4725,    945,     63,     1;

%e   1,  84, 1764, 13230,  13230,   1764,    84,    1;

%e   1, 108, 3024, 31752, 142884,  31752,  3024,  108,   1;

%e   1, 135, 4860, 68040, 428652, 428652, 68040, 4860, 135, 1;

%t T[n_,k_]:= (Binomial[n-1, k-1]*Binomial[n, k-1]/k)*If[Floor[n/2]>k-1, 3^(k-1), 3^(n-k)];

%t Table[T[n,k], {n,12}, {k,n}]//Flatten

%o (Magma)

%o function T(n,k)

%o   if Floor(n/2) gt k-1 then return (1/n)*Binomial(n,k)*Binomial(n,k-1)*3^(k-1);

%o   else return (1/n)*Binomial(n,k)*Binomial(n,k-1)*3^(n-k);

%o   end if; return T;

%o end function;

%o [T(n,k): k in [1..n], n in [1..12]]; // _G. C. Greubel_, Nov 26 2021

%o (Sage)

%o def A174346(n,k): return (1/n)*binomial(n,k)*binomial(n,k-1)*( 3^(k-1) if ((n//2)>k-1) else 3^(n-k) )

%o flatten([[A174346(n,k) for k in (1..n)] for n in (1..12)]) # _G. C. Greubel_, Nov 26 2021

%Y Cf. A081582.

%K nonn,tabl

%O 1,5

%A _Roger L. Bagula_, Mar 16 2010

%E Edited by _G. C. Greubel_, Nov 26 2021