A202409 - OEIS

%I #13 Jul 02 2013 09:08:44

%S 1,4,4,9,36,9,16,168,168,16,25,550,1400,550,25,36,1440,7500,7500,1440,

%T 36,49,3234,30135,61250,30135,3234,49,64,6496,98784,356720,356720,

%U 98784,6496,64,81,11988,278208,1629936,2889432,1629936,278208,11988,81

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

%C Let a meander be defined as in the link and m = 3. Then T(n,k) counts the invertible meanders of length m(n+1) built from arcs with central angle 360/m whose binary representation have mk '1's.

%H Peter Luschny, <a href="http://oeis.org/wiki/User:Peter_Luschny/Meander">Meanders</a>.

%e [1]                1

%e [2]               4, 4

%e [3]             9, 36, 9

%e [4]         16, 168, 168, 16

%e [5]      25, 550, 1400, 550, 25

%e [6]  36, 1440, 7500, 7500, 1440, 36

%e T(2,1) = 4 because the invertible meanders of length 9 and central angle 120 degree which have three '1's in their binary representation are {100100100, 100011000, 110001000, 111000000}.

%p A202409 := (n,k) -> k*binomial(n,k)^3*(n^2+n-k*n-k+k^2)/((n-k+1)^2*n);

%p seq(print(seq(A202409(n,k),k=1..n)),n=1..6);

%t t[n_, k_] := k*Binomial[n, k]^3*(n^2 + n - k*n - k + k^2)/((n - k + 1)^2*n); Table[t[n, k], {n, 1, 9}, {k, 1, n}] // Flatten (* _Jean-François Alcover_, Jul 02 2013 *)

%Y Row sums: A201640. Cf. A132812.

%K nonn,tabl

%O 1,2

%A _Peter Luschny_ and _Susanne Wienand_, Dec 19 2011