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 1-4 of 4 results.

A120101 Triangle T(n,k) = lcm(1,...,2*n+2)/((k+1)*binomial(2*k+2,k+1)).

Original entry on oeis.org

1, 6, 1, 30, 5, 1, 420, 70, 14, 3, 1260, 210, 42, 9, 2, 13860, 2310, 462, 99, 22, 5, 180180, 30030, 6006, 1287, 286, 65, 15, 360360, 60060, 12012, 2574, 572, 130, 30, 7, 6126120, 1021020, 204204, 43758, 9724, 2210, 510, 119, 28, 116396280, 19399380, 3879876, 831402, 184756, 41990, 9690, 2261, 532, 126
Offset: 0

Views

Author

Paul Barry, Jun 09 2006

Keywords

Comments

The rows give the coefficients of polynomials arising in the integration of x^(2m)/sqrt(4-x^2), m >= 0.

Examples

			Triangle begins:
       1;
       6,     1;
      30,     5,     1;
     420,    70,    14,    3;
    1260,   210,    42,    9,   2;
   13860,  2310,   462,   99,  22,   5;
  180180, 30030,  6006, 1287, 286,  65, 15;
  360360, 60060, 12012, 2574, 572, 130, 30, 7;

Links

Crossrefs

First column is A119634. Second column is A051538. Main diagonal is A068553. Subdiagonal is A119636. Inverse is A120113. Row sums are A120106. Diagonal sums are A120107.
Cf. A000984, A003418.

Programs

  • GAP
    Flat(List([0..9],n->List([0..n],k->Lcm(List([1..2*n+2],i->i))/((k+1)*Binomial(2*k+2,k+1))))); # Muniru A Asiru, Feb 26 2019
  • Magma
    [Lcm([1..2*n+2])/((k+1)*(k+2)*Catalan(k+1)): k in [0..n], n in [0..12]]; // G. C. Greubel, May 03 2023
  • Maple
    T:=(n,k)-> ilcm(seq(q,q=1..2*n+2))/((k+1)*binomial(2*k+2,k+1)): seq(seq(T(n,k),k=0..n),n=0..9); # Muniru A Asiru, Feb 26 2019
  • Mathematica
    Table[LCM@@Range[2*n+2]/((k+1)*Binomial[2*k+2,k+1]), {n,0,12}, {k,0, n}]//Flatten (* G. C. Greubel, May 03 2023 *)
  • SageMath
    def A120101(n,k):
    return lcm(range(1,2*n+3))/((k+1)*(k+2)*catalan_number(k+1))
flatten([[A120101(n,k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, May 03 2023

Formula

Number triangle T(n,k) = [k<=n] * lcm(1,...,2n+2)/((k+1)*binomial(2k+2, k+1)).

A025555 Least common multiple (or LCM) of first n positive triangular numbers (A000217).

Original entry on oeis.org

1, 3, 6, 30, 30, 210, 420, 1260, 1260, 13860, 13860, 180180, 180180, 180180, 360360, 6126120, 6126120, 116396280, 116396280, 116396280, 116396280, 2677114440, 2677114440, 13385572200, 13385572200, 40156716600, 40156716600
Offset: 1

Views

Author

Clark Kimberling and Asher Auel

Keywords

Examples

			a(5) = lcm{1, 3, 6, 10, 15} = 30.

Links

Crossrefs

Cf. A051543, A051538.

Programs

  • Haskell
    a025555 n = a025555_list !! (n-1)
a025555_list = scanl1 lcm $ tail a000217_list
-- Reinhard Zumkeller, Nov 22 2013
  • Maple
    HalfFarey := proc (n) local a,b,c,d,k,s; if n<2 then RETURN([1]) fi; a:=0; b:=1; c:=1; d:=n; s:=NULL; do k := iquo(n+b,d); a,b,c,d := c, d, k*c-a, k*d-b; if b < 2*a then break fi; s := s, a/b od; [s] end:
A025555 := proc(n) local r; HalfFarey(n+1); subsop(nops(%) = NULL,%); mul(2*sin(Pi*r),r = %)^2 end: seq(round(evalf(A025555(i))),i=1..27); # Peter Luschny, Jun 09 2011
  • Mathematica
    nn=30;With[{trnos=Accumulate[Range[nn]]},Table[LCM@@Take[trnos,n], {n,nn}]] (* Harvey P. Dale, Oct 21 2011 *)
f[x_] := x + 1; a[1] = f[1]; a[n_] := LCM[f[n], a[n - 1]]; Array[a, 30]/2 (* Robert G. Wilson v, Jan 04 2013 *)
  • PARI
    S=1;for(n=1,20,S=lcm(S,n*(n+1)/2);print1(S,",")) \\ Edward Jiang, Sep 08 2014

Formula

a(n) = A003418(n+1)/2. - Matthew Vandermast, Jun 04 2012

Extensions

Corrected by James Sellers
Definition rendered more precisely by Reinhard Zumkeller, Nov 22 2013

A051542 Quotients of consecutive values of LCM {b(1),...,b(n)}, b() = A000330.

Original entry on oeis.org

5, 14, 3, 11, 13, 2, 17, 19, 1, 23, 5, 3, 29, 62, 1, 1, 37, 1, 41, 43, 1, 47, 7, 1, 53, 1, 1, 59, 61, 2, 1, 67, 1, 71, 73, 1, 1, 79, 3, 83, 1, 1, 89, 1, 1, 1, 97, 1, 101, 103, 1, 107, 109, 1, 113, 1, 1, 1, 11, 1, 5, 254, 1, 131, 1, 1, 137, 139, 1, 1, 1, 1, 149, 151, 1, 1, 157, 1, 1
Offset: 1

Views

Author

Asher Auel

Keywords

Examples

			a(3) = A051538(4)/A051538(3) = 210/70 = 3

Links

Crossrefs

Cf. A051538,A051543.

Programs

  • Haskell
    a051542 n = a051542_list !! (n-1)
a051542_list = zipWith div (tail a051538_list) a051538_list
-- Reinhard Zumkeller, Mar 12 2014

Formula

a(n) = A051538(n+1)/A051538(n)

Extensions

Corrected and extended by James Sellers
Example fixed by Reinhard Zumkeller, Mar 12 2014

A120105 Number triangle T(n,k) = lcm(1,..,2*n+2)/lcm(1,..,2*k+2).

Original entry on oeis.org

1, 6, 1, 30, 5, 1, 420, 70, 14, 1, 1260, 210, 42, 3, 1, 13860, 2310, 462, 33, 11, 1, 180180, 30030, 6006, 429, 143, 13, 1, 360360, 60060, 12012, 858, 286, 26, 2, 1, 6126120, 1021020, 204204, 14586, 4862, 442, 34, 17, 1, 116396280, 19399380, 3879876, 277134, 92378, 8398, 646, 323, 19, 1
Offset: 0

Views

Author

Paul Barry, Jun 09 2006

Keywords

Examples

			Triangle begins:
       1;
       6,     1;
      30,     5,    1;
     420,    70,   14,   1;
    1260,   210,   42,   3,   1;
   13860,  2310,  462,  33,  11,   1;
  180180, 30030, 6006, 429, 143,  13,  1;

Links

Crossrefs

First column is A119634. Second column is A051538. Inverse is A120111.
Cf. A120106, A120107.

Programs

  • GAP
    Flat(List([0..9],n->List([0..n],k->Lcm(List([1..2*n+2],i->i))/Lcm(List([1..2*k+2],i->i))))); # Muniru A Asiru, Feb 26 2019
  • Magma
    [Lcm([1..2*n+2])/Lcm([1..2*k+2]): k in [0..n], n in [0..12]]; // G. C. Greubel, May 04 2023
  • Maple
    T:= (n,k)-> ilcm(seq(q,q=1..2*n+2))/ilcm(seq(r,r=1..2*k+2)):
seq(seq(T(n,k),k=0..n),n=0..9); # Muniru A Asiru, Feb 26 2019
  • Mathematica
    T[n_, k_]:= LCM@@Range[2*n+2]/(LCM@@Range[2*k+2]);
Table[T[n,k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, May 04 2023 *)
  • SageMath
    def f(n): return lcm(range(1,2*n+3))
def A120105(n,k):
    return f(n)//f(k)
flatten([[A120105(n,k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, May 04 2023

Formula

Number triangle T(n,k) = [k<=n] + lcm(1,..,2n+2)/lcm(1,..,2k+2).
From G. C. Greubel, May 04 2023: (Start)
Sum_{k=0..n} T(n, k) = A120106(n).
Sum_{k=0..floor(n/2)} T(n-k, k) = A120107(n). (End)
Showing 1-4 of 4 results.

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