A093527 -id:A093527 - 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

A098505 Numerators in inverse of a Catalan scaled binomial matrix.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 3, 3, 1, 1, 2, 3, 2, 1, 1, 5, 5, 5, 5, 1, 1, 1, 5, 5, 5, 1, 1, 1, 7, 7, 35, 35, 7, 7, 1, 1, 4, 14, 28, 7, 28, 14, 4, 1, 1, 9, 18, 42, 63, 63, 42, 18, 9, 1, 1, 5, 45, 30, 105, 63, 105, 30, 45, 5, 1, 1, 11, 55, 165, 165, 33, 33, 165, 165, 55, 11, 1, 1, 3, 33, 55, 495, 198

Offset: 0

Paul Barry, Sep 11 2004

Row sums are A098506. Diagonal sums are A098507. Second column is A093527. Third column is A098508. Numerators in the inverse of the signed version of A098474, defined by T(n,k)=(-1)^(n-k)binomial(2k,k)binomial(n,k)/(k+1)

			Rows begin:
  1;
  1,1;
  1,1,1;
  1,3,3,1;
  1,2,3,2,1;
  1,5,5,5,5,1;
  ...

Paolo Xausa, Table of n, a(n) for n = 0..11475 (rows 0..150 of triangle, flattened).

Cf. A098474, A098506, A098507, A098508, A093527.

Table[Numerator[(n + 1)*Binomial[n, k]/Binomial[2*n, n]], {n, 0, 15}, {k, 0, n}] (* Paolo Xausa, Aug 30 2024 *)

T(n, k) = numerator((n+1)*binomial(n, k)/binomial(2n, n)).

A093528 Numerators of odd raw moments in the distribution of line lengths for lines picked at random in the unit disk.

Original entry on oeis.org

128, 2048, 16384, 524288, 4194304, 67108864, 536870912, 34359738368, 274877906944, 4398046511104, 35184372088832, 1125899906842624, 9007199254740992, 144115188075855872, 1152921504606846976

Offset: 1

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, A093526, A093527, A093529.

a(k) = Numerator[(2*Gamma[3 + n])/((2 + n)*Gamma[2 + n/2]*Gamma[3 + n/2])] for n = 2k-1.

A093529 Pi*denominators of odd raw moments in the distribution of line lengths for lines picked at random in the unit disk.

Original entry on oeis.org

45, 525, 2205, 31185, 99099, 585585, 1640925, 35334585, 92147055, 468495027, 1166167275, 11408158125, 27484885575, 130734984825, 307452619485, 11455089532425, 26442675480375, 121132637200575, 275520749478975

Offset: 1

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, A093526, A093527, A093528.

a(k) = Pi*Denominator[(2*Gamma[3 + n])/((2 + n)*Gamma[2 + n/2]*Gamma[3 + n/2])] for n = 2k-1.

A098512 Second column and subdiagonal of number triangle A098509.

Original entry on oeis.org

1, 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

Paul Barry, Sep 11 2004

Cf. A093526, A093527, A000108.

C := n -> binomial(2*n,n)/(n+1):
A098512 := n -> C(n)/igcd(n,C(n)): # Peter Luschny, Oct 06 2011

a(n) = denominator(n(n+1)/binomial(2n, n)).
a(n) = denominator(n/C(n)). - Paul Barry, Nov 17 2004
a(n) = C(n) / gcd(n, C(n)). - Peter Luschny, Oct 06 2011

A195686 a(n) = C(2n,n) / gcd(n,C(2n,n)).

Original entry on oeis.org

1, 2, 3, 20, 35, 252, 154, 3432, 6435, 48620, 92378, 705432, 676039, 10400600, 20058300, 10341168, 300540195, 2333606220, 1512522550, 35345263800, 6892326441, 179419291480, 1052049481860, 8233430727600, 2687300306925, 126410606437752, 247959266474052

Offset: 0

Peter Luschny, Oct 06 2011

nonn

T. D. Noe, Table of n, a(n) for n = 0..400

Cf. A093526, A093527, A145429.

A195686  := n -> binomial(2*n,n)/igcd(n,binomial(2*n,n));

a[n_] := Numerator[Binomial[2n,n]/n]; Join[{1}, Table[a[n], {n, 100}]] (* Enrique Pérez Herrero, Mar 26 2012 *)

A093526(n) = a(n+1)/(n+2).
a(n) = numerator(C(2n,n)/n). - Enrique Pérez Herrero, Mar 26 2012
Sum_{n>=0} A093527(n)/a(n+1) = Sum_{n>=1} n/binomial(2*n,n) = 2/3 + 2*Pi/(9*sqrt(3)) (A145429). - Amiram Eldar, Jan 26 2022
a(n) = numerator((n + 1)*binomial(2*n+1, n)/(n*(2*n + 1))) for n > 0. - Stefano Spezia, Aug 06 2022

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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A098505 Numerators in inverse of a Catalan scaled binomial matrix.

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A093528 Numerators of odd raw moments in the distribution of line lengths for lines picked at random in the unit disk.

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A093529 Pi*denominators of odd raw moments in the distribution of line lengths for lines picked at random in the unit disk.

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A098512 Second column and subdiagonal of number triangle A098509.

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A195686 a(n) = C(2*n,n) / gcd(n,C(2*n,n)).

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A195686 a(n) = C(2n,n) / gcd(n,C(2n,n)).