cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-6 of 6 results.

A093412 Triangle read by rows: a(n, k) is the numerator of (n + (n-1) + ... + (n-k+1))/(1 + 2 + ... + k), 0 < k <= n.

Original entry on oeis.org

1, 2, 1, 3, 5, 1, 4, 7, 3, 1, 5, 3, 2, 7, 1, 6, 11, 5, 9, 4, 1, 7, 13, 3, 11, 5, 9, 1, 8, 5, 7, 13, 2, 11, 5, 1, 9, 17, 4, 3, 7, 13, 3, 11, 1, 10, 19, 9, 17, 8, 15, 7, 13, 6, 1, 11, 7, 5, 19, 3, 17, 2, 5, 7, 13, 1, 12, 23, 11, 21, 10, 19, 9, 17, 8, 15, 7, 1, 13, 25, 6, 23, 11, 3, 5, 19, 9, 17, 4
Offset: 1

Views

Author

Amarnath Murthy, Mar 30 2004

Keywords

Comments

A093415 gives the corresponding denominators.

Examples

			Triangle a(n,k) (with rows n >= 1 and columns k >= 1) begins as follows:
  1;
  2,  1;
  3,  5, 1;
  4,  7, 3,  1;
  5,  3, 2,  7, 1;
  6, 11, 5,  9, 4,  1;
  7, 13, 3, 11, 5,  9, 1;
  8,  5, 7, 13, 2, 11, 5,  1;
  9, 17, 4,  3, 7, 13, 3, 11, 1;
... - _Petros Hadjicostas_, Oct 20 2019

Crossrefs

Cf. A093413, A093414, A093415.

Formula

a(n, k) = (2n+1-k)/gcd(2n+1-k, k+1).

Extensions

Edited and extended by David Wasserman, Feb 01 2006

A093415 Triangle read by rows: a(n, k) is the denominator of (n + (n-1) + ... + (n-k+1))/(1 + 2 + ... + k), 0 < k <= n.

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 3, 2, 1, 1, 1, 1, 5, 1, 1, 3, 2, 5, 3, 1, 1, 3, 1, 5, 3, 7, 1, 1, 1, 2, 5, 1, 7, 4, 1, 1, 3, 1, 1, 3, 7, 2, 9, 1, 1, 3, 2, 5, 3, 7, 4, 9, 5, 1, 1, 1, 1, 5, 1, 7, 1, 3, 5, 11, 1, 1, 3, 2, 5, 3, 7, 4, 9, 5, 11, 6, 1, 1, 3, 1, 5, 3, 1, 2, 9, 5, 11, 3, 13, 1, 1, 1, 2, 1, 1, 7, 4, 3, 1, 11, 2
Offset: 1

Views

Author

Amarnath Murthy, Mar 30 2004

Keywords

Comments

A093412 gives the corresponding numerators.
A109613(n+1) - 2 = 2*floor((n+1)/2) - 1 is the largest number in row n. [Corrected by Petros Hadjicostas, Oct 20 2019]

Examples

			Triangle a(n,k) (with rows n >= 1 and columns k >= 1) begins as follows:
  1;
  1, 1;
  1, 3, 1;
  1, 3, 2, 1;
  1, 1, 1, 5, 1;
  1, 3, 2, 5, 3, 1;
  1, 3, 1, 5, 3, 7, 1;
  1, 1, 2, 5, 1, 7, 4, 1;
  1, 3, 1, 1, 3, 7, 2, 9, 1;
... - _Petros Hadjicostas_, Oct 20 2019

Crossrefs

Cf. A109613, A093412, A093413, A093414, A093417.

Formula

a(n, k) = (k+1)/gcd(2n+2, k+1).

Extensions

Edited and extended by David Wasserman, Feb 01 2006

A093414 Row sums of A093412.

Original entry on oeis.org

1, 3, 9, 15, 18, 36, 49, 52, 68, 105, 90, 153, 151, 136, 225, 276, 217, 351, 300, 297, 416, 528, 402, 538, 598, 563, 639, 861, 547, 990, 961, 808, 1061, 945, 913, 1431, 1342, 1158, 1268, 1770, 1158, 1953, 1704, 1351, 2003, 2346, 1698, 2268, 2051, 2047
Offset: 1

Views

Author

Amarnath Murthy, Mar 30 2004

Keywords

Crossrefs

Cf. A093412, A093413, A093415, A093417.

Extensions

Edited and extended by David Wasserman, Feb 01 2006

A093419 Denominators of row sums in triangle described in A093412.

Original entry on oeis.org

1, 1, 3, 6, 5, 10, 35, 140, 126, 1260, 1155, 13860, 12870, 12012, 45045, 360360, 340340, 2042040, 1939938, 369512, 117572, 2586584, 7436429, 178474296, 171609900, 1487285800, 1434168450, 40156716600, 38818159380, 1164544781400
Offset: 1

Views

Author

Amarnath Murthy, Mar 30 2004

Keywords

Crossrefs

Essentially the same as A096620.
Cf. A093412, A093413, A093414, A093415, A093417, A093418.

Programs

  • Mathematica
    Do[s = i = n; j = 1; x = i; y = j; While[x/y != 1, i--; j++; x += i; y += j; s += x/y]; Print[Denominator[s]], {n, 1, 30}] (* Ryan Propper, Aug 16 2005 *)

Extensions

Corrected and extended by Ryan Propper, Aug 16 2005

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

Views

Author

Amarnath Murthy, Mar 30 2004

Keywords

Examples

			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,

Links

Crossrefs

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

Programs

  • Magma
    /* 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
  • Maple
    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
  • Mathematica
    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 *)
  • PARI
    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

Formula

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

Extensions

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

Views

Author

Amarnath Murthy, Mar 30 2004

Keywords

Examples

			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;
  ...

Links

Crossrefs

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

Programs

  • Magma
    /* 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
  • Maple
    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
  • Mathematica
    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 *)
  • PARI
    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

Formula

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

Extensions

More terms from R. J. Mathar, Apr 28 2007
Better definition from Omar E. Pol, Jan 10 2009
Showing 1-6 of 6 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.