A093420 -id:A093420 - cp's OEIS frontend

A090586 Denominator of Sum/Product of first n numbers.

1, 2, 1, 12, 8, 240, 180, 1120, 8064, 725760, 604800, 79833600, 68428800, 830269440, 10897286400, 2615348736000, 2324754432000, 711374856192000, 640237370572800, 11585247657984000, 221172909834240000, 102181884343418880000, 93666727314800640000

Offset: 1

Reinhard Zumkeller, Dec 03 2003

a(n) = A000142(n)/A069268(n);
a(p-1) = 2*A000142(p-2) for odd primes.
a(n) = A060593((n - 1)/2) for odd n. - Gregory Gerard Wojnar, Jun 10 2021

Robert Israel, Table of n, a(n) for n = 1..451

Cf. A090585 (numerator), A060593.

Main diagonal of A093420.

[Denominator(n + 1) / (2*Factorial(n - 1)): n in [1..30]]; // Vincenzo Librandi, Oct 15 2018

seq(denom((n+1)/(2*(n-1)!)),n=1..25); # Robert Israel, Oct 14 2018

Table[Denominator[(n + 1) / (2 (n - 1)!)], {n, 25}] (* Vincenzo Librandi, Oct 15 2018 *)

A093421 Triangle read by rows: T(n,k) is the denominator of f(n, k) = (Product_{i = 0..k-1} (n-i))/(Sum_{i = 1..k} i) for 1 <= k <= n.

Original entry on oeis.org

1, 1, 3, 1, 1, 1, 1, 1, 1, 5, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 7, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 5, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 11, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 13, 1, 1, 1, 1, 1, 7, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 5, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 1

Amarnath Murthy, Mar 30 2004

			Triangle T(n,k) (with rows n >= 1 and columns k >= 1) begins as follows:
  1;
  1, 3;
  1, 1, 1;
  1, 1, 1, 5;
  1, 3, 1, 1, 1;
  1, 1, 1, 1, 1, 7;
  1, 1, 1, 1, 1, 1, 1;
  1, 3, 1, 1, 1, 1, 1, 1;
  1, 1, 1, 5, 1, 1, 1, 1, 1;
  1, 1, 1, 1, 1, 1, 1, 1, 1, 11;
  ...

Cf. A090585, A090586, A093415, A093420 (numerators), A093423.

T(n,n) = denominator(f(n, n)) = denominator(2*(n-1)!/(n+1)).

Edited and extended by David Wasserman, Aug 29 2006

A093422 In the triangle shown below the n-th row contains n rational numbers n/1, {n(n-1)}/{n +(n-1)}, {(n)(n-1)(n-2)}/{n +(n-1)+(n-2)}, ..., the last term being 2(n-1)!/(n+1). Sequence gives the numerators in each row.

Original entry on oeis.org

1, 2, 2, 3, 6, 1, 4, 12, 8, 12, 5, 20, 5, 60, 8, 6, 30, 8, 20, 36, 240, 7, 42, 35, 420, 504, 560, 180, 8, 56, 16, 840, 224, 6720, 1152, 1120, 9, 72, 21, 504, 432, 20160, 4320, 90720, 8064, 10, 90, 80, 2520, 756, 3360, 86400, 453600, 67200, 725760, 11, 110, 33, 3960

Offset: 1

Amarnath Murthy, Mar 30 2004

			Triangle of fractions starts
  1,
  2, 2/3,
  3, 6/5, 1,
  4, 12/7, 8/3, 12/5,
  5, 20/9, 5, 60/7, 8,
  6, 30/11, 8, 20, 36, 240/7,
  7, 42/13, 35/3, 420/11, 504/5, 560/3, 180,
  8, 56/15, 16, 840/13, 224, 6720/11, 1152, 1120,
  9, 72/17, 21, 504/5, 432, 20160/13, 4320, 90720/11, 8064,

G. C. Greubel, Rows n=0..100 of triangle, flattened

Cf. A093412, A093413, A093414, A093415, A093417, A093418, A093419, A093420, A093421, A093423.

/* as a triangle */ [[k le 1 select n else Numerator(2*Binomial(n,k)*Factorial(k-1)/(2*n-k+1)): k in [1..n]]: n in [1..10]]; // G. C. Greubel, Sep 01 2018

A09342x := proc(n,m) local i,N,D ; N := n ; if m = 1 then D := 1 ; else D := n ; end ; for i from 1 to m-1 do N := N*(n-i) ; D := D+n-i ; od ; simplify(N/D) ; end: A093422 := proc(n,m) numer(A09342x(n,m)) ; end: for n from 1 to 12 do for m from 1 to n do printf("%d, ",A093422(n,m)) ; od ; od ; # R. J. Mathar, Apr 28 2007

Table[If[k == 1, n, Numerator[2*Binomial[n,k]*(k-1)!/(2*n-k+1)]], {n,1,30}, {k,1,n}]//Flatten (* G. C. Greubel, Sep 01 2018 *)

for(n=1,10, for(k=1,n, print1(if(k==1, n, denominator(2*binomial(n,k)*(k-1)!/(2*n-k+1))), ", "))) \\ G. C. Greubel, Sep 01 2018

A093422(n,m)/A093423(n,m) = 2*binomial(n,m)*(m-1)!/(2*n-m+1) for 2 <= m < n. A093422(n,1)/A093423(n,1)= n. - R. J. Mathar, Apr 28 2007

More terms from R. J. Mathar, Apr 28 2007

A093423 Consider the triangle whose first part is shown as an example in the entry A093422. If the n-th term of the triangle read by rows is a fraction then a(n) is the denominator of the fraction, otherwise a(n)=1.

Original entry on oeis.org

1, 1, 3, 1, 5, 1, 1, 7, 3, 5, 1, 9, 1, 7, 1, 1, 11, 1, 1, 1, 7, 1, 13, 3, 11, 5, 3, 1, 1, 15, 1, 13, 1, 11, 1, 1, 1, 17, 1, 5, 1, 13, 1, 11, 1, 1, 19, 3, 17, 1, 1, 7, 13, 1, 11, 1, 21, 1, 19, 1, 17, 1, 1, 1, 13, 1, 1, 23, 1, 7, 5, 19, 1, 17, 1, 1, 1, 13

Offset: 1

Amarnath Murthy, Mar 30 2004

			Triangle begins:
  1;
  1,  3;
  1,  5,  1;
  1,  7,  3,  5;
  1,  9,  1,  7,  1;
  1, 11,  1,  1,  1,  7;
  1, 13,  3, 11,  5,  3,  1;
  1, 15,  1, 13,  1, 11,  1,  1;
  ...

G. C. Greubel, Rows n=1..100 of triangle, flattened

Cf. A093412, A093413, A093414, A093415, A093417, A093418, A093419, A093420, A093421, A093422.

/* as a triangle */ [[Denominator(2*Binomial(n,k)*Factorial(k-1)/(2*n-k+1)): k in [1..n]]: n in [1..30]]; // G. C. Greubel, Sep 01 2018

A09342x := proc(n,m) local a,i,N,D ; N := n ; if m = 1 then D := 1 ; else D := n ; end ; for i from 1 to m-1 do N := N*(n-i) ; D := D+n-i ; od ; simplify(N/D) ; end: A093423 := proc(n,m) denom(A09342x(n,m)) ; end: for n from 1 to 12 do for m from 1 to n do printf("%d, ",A093423(n,m)) ; od ; od ; # R. J. Mathar, Apr 28 2007

Table[Denominator[2*Binomial[n,k]*(k-1)!/(2*n-k+1)], {n,1,30}, {k,1,n}]//Flatten (* G. C. Greubel, Sep 01 2018 *)

for(n=1,10, for(k=1,n, print1(denominator(2*binomial(n,k)*(k-1)!/(2*n-k+1)), ", "))) \\ G. C. Greubel, Sep 01 2018

A093422(n,m)/A093423(n,m) = 2*binomial(n,m)*(m-1)!/(2*n-m+1) for 2 <= m < n. A093422(n,1)/A093423(n,1)= n. - R. J. Mathar, Apr 28 2007

More terms from R. J. Mathar, Apr 28 2007

Better definition from Omar E. Pol, Jan 10 2009

cp's OEIS Frontend

A090586 Denominator of Sum/Product of first n numbers.

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A093421 Triangle read by rows: T(n,k) is the denominator of f(n, k) = (Product_{i = 0..k-1} (n-i))/(Sum_{i = 1..k} i) for 1 <= k <= n.

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A093422 In the triangle shown below the n-th row contains n rational numbers n/1, {n*(n-1)}/{n +(n-1)}, {(n)*(n-1)*(n-2)}/{n +(n-1)+(n-2)}, ..., the last term being 2*(n-1)!/(n+1). Sequence gives the numerators in each row.

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Magma

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A093423 Consider the triangle whose first part is shown as an example in the entry A093422. If the n-th term of the triangle read by rows is a fraction then a(n) is the denominator of the fraction, otherwise a(n)=1.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

Formula

Extensions

A093422 In the triangle shown below the n-th row contains n rational numbers n/1, {n(n-1)}/{n +(n-1)}, {(n)(n-1)(n-2)}/{n +(n-1)+(n-2)}, ..., the last term being 2(n-1)!/(n+1). Sequence gives the numerators in each row.