A093423 -id:A093423 - cp's OEIS frontend

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.

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

cp's OEIS Frontend

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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cp's OEIS Frontend

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.

Views

Author

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Examples

Crossrefs

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.

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.