A098512 -id:A098512 - cp's OEIS frontend

A093526 Numerators of even raw moments in the distribution of line lengths for lines picked at random in the unit disk.

1, 1, 5, 7, 42, 22, 429, 715, 4862, 8398, 58786, 52003, 742900, 1337220, 646323, 17678835, 129644790, 79606450, 1767263190, 328206021, 8155422340, 45741281820, 343059613650, 107492012277, 4861946401452, 9183676536076

Offset: 0

Eric W. Weisstein, Mar 30 2004

			1, 128/(45*Pi), 1, 2048/(525*Pi), 5/3, 16384/(2205*Pi), ...

Eric Weisstein's World of Mathematics, Disk Line Picking

Cf. A093070, A093527, A093528, A093529.

Cf. A098512, A000108.

A195686 := n -> binomial(2*n,n) / igcd(n,binomial(2*n,n));
A093526 := n -> A195686(n+1)/(n+2); # Peter Luschny, Oct 06 2011

a[n_] := Binomial[2(n+1), n+1]/((n+2) GCD[n+1, Binomial[2(n+1), n+1]]);
Table[a[n], {n, 0, 25}] (* Jean-François Alcover, Jul 30 2018, after Peter Luschny *)

a(n) = denominator((n+1)*(n+2)/binomial(2*n+2, n+1)); \\ Michel Marcus, Jul 30 2018

a(k) = Numerator[(2*Gamma[3 + n])/((2 + n)*Gamma[2 + n/2]*Gamma[3 + n/2])] for n = 2k.
a(n) = denominator((n+1)/C(n+1)). - Paul Barry, Nov 17 2004
a(n) = A195686(n+1) / (n+2). - Peter Luschny, Oct 06 2011

A098509 Denominators of the inverse of a Catalan scaled binomial matrix.

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 5, 5, 5, 5, 14, 7, 7, 7, 14, 42, 42, 21, 21, 42, 42, 132, 22, 44, 33, 44, 22, 132, 429, 429, 143, 429, 429, 143, 429, 429, 1430, 715, 715, 715, 143, 715, 715, 715, 1430, 4862, 4862, 2431, 2431, 2431, 2431, 2431, 2431, 4862, 4862, 16796, 8398

Offset: 0

Paul Barry, Sep 11 2004

First column and main diagonal are A000108. Row sums are A098510. Diagonal sums are A098511. Second column is A098512.

			Rows begin
1;
1,1;
2,1,2;
5,5,5,5;
14,7,7,7,14;
...

Cf. A098505, A098474.

T[n_, k_] := Denominator[(n + 1)*(Binomial[n, k]/Binomial[2*n, n])];
Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jan 07 2018 *)

Triangle T(n, k)=denominator((n+1)binomial(n, k)/binomial(2n, n))

cp's OEIS Frontend

A093526 Numerators of even raw moments in the distribution of line lengths for lines picked at random in the unit disk.

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