A036064 Triangle of flag coefficients [ n k ] (numerators of rational parts).

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

Offset

0,8

References

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

Links

Table of n, a(n) for n=0..79.

Formula

[ 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).

Example

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; ...

Mathematica

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 *)

Crossrefs

Cf. A036065.

Sequence in context: A258140 A140408 A047080 * A352744 A347967 A090706

Adjacent sequences: A036061 A036062 A036063 * A036065 A036066 A036067

Keyword

nonn,easy,tabl,nice,frac

Author

N. J. A. Sloane

Extensions

More terms from Naohiro Nomoto, Jun 20 2001

Status

approved