A036064 - OEIS

%I #15 Oct 22 2020 18:51:52

%S 1,1,1,1,1,1,1,2,2,1,1,3,3,3,1,1,8,4,4,8,1,1,15,5,15,5,15,1,1,16,6,8,

%T 8,6,16,1,1,35,7,105,35,105,7,35,1,1,128,8,64,16,16,64,8,128,1,1,315,

%U 9,315,21,945,21,315,9,315,1,1,256,10,128,80,32,32,80,128,10,256,1,1,693

%N Triangle of flag coefficients [ n k ] (numerators of rational parts).

%D D. A. Klain and G.-C. Rota, Introduction to Geometric Probability, Cambridge, p. 69.

%F [ 2m 2k ] = C(2m, 2k)/C(m, k); [ 2m 2k+1 ] = Pi(2m)!/4^m*k!m!(m-k-1)!; [ 2m+1 2k ] = 4^k*C(m, k)/C(2k, k); [ 2m+1 2k+1 ] = 4^(m-k)*C(m, m-k)/C(2m-2k, m-k).

%e 1; 1 1; 1 Pi/2 1; 1 2 2 1; 1 3Pi/4 3 3Pi/4 1; 1 8/3 4 4 8/3 1; ...

%t f[m_?EvenQ, k_?EvenQ] := Binomial[m, k]/Binomial[m/2, k/2]; f[m_?EvenQ, k_?OddQ] := (Pi*m!)/(2^m*(((1/2)*(k - 1))!*(-1 + (1 - k)/2 + m/2)!*(m/2)!)); f[m_?OddQ, k_?EvenQ] := (2^k*Binomial[(1/2)*(m - 1), k/2])/ Binomial[k, k/2]; f[m_?OddQ, k_?OddQ] := (4^((1 - k)/2 + (1/2)*(m - 1))* Binomial[(1/2)*(m - 1), (1 - k)/2 + (1/2)*(m - 1)]) / Binomial[m - k , (1 - k)/2 + (1/2)*(m - 1)]; A036064 = Numerator[ Flatten[ Table[ f[m, k], {m, 0, 12}, {k, 0, m}] /. Pi -> 1]](* _Jean-François Alcover_, May 10 2012, from formula *)

%Y Cf. A036065.

%K nonn,easy,tabl,nice,frac

%O 0,8

%A _N. J. A. Sloane_

%E More terms from _Naohiro Nomoto_, Jun 20 2001