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 1-3 of 3 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)).

A120107 a(n) = Sum_{k=0..floor(n/2)} lcm(1,...,2*(n-k)+2)/lcm(1,...,2*k+2).

Original entry on oeis.org

1, 6, 31, 425, 1331, 14084, 182533, 390855, 6192220, 117429752, 136000866, 2700408581, 13835919839, 42477252404, 1171690228133, 72397239805085, 84274330442804, 86644937313210, 2686078920033439, 3119346038772923
Offset: 0

Views

Author

Paul Barry, Jun 09 2006

Keywords

Comments

Diagonal sums of number triangle A120101.

Links

Crossrefs

Cf. A120101, A120105, A120106.

Programs

  • GAP
    List([0..20],n->Sum([0..Int(n/2)],k->Lcm(List([1..2*(n-k)+2],i->i))/Lcm(List([1..2*k+2],i->i)))); # Muniru A Asiru, Mar 03 2019
  • Magma
    A120105:= func< n,k | Lcm([1..2*n+2])/Lcm([1..2*k+2]) >;
[(&+[A120105(n-k,k): k in [0..Floor(n/2)]]): n in [0..50]]; // G. C. Greubel, May 04 2023
  • Mathematica
    A120105[n_, k_]:= LCM@@Range[2*n+2]/(LCM@@Range[2*k+2]);
A120107[n_]:= Sum[A120105[n-k, k], {k, 0, Floor[n/2]}];
Table[A120107[n], {n,0,50}] (* G. C. Greubel, May 04 2023 *)
  • PARI
    a(n) = sum(k=0, n\2, lcm([1..2*(n-k)+2])/lcm([1..2*k+2])); \\ Michel Marcus, Mar 04 2019
  • SageMath
    def f(n): return lcm(range(1,2*n+3))
def A120107(n):
    return sum(f(n-k)/f(k) for k in range(1+(n//2)))
[A120107(n) for n in range(51)] # G. C. Greubel, May 04 2023

Formula

a(n) = Sum_{k=0..floor(n/2)} A120105(n-k, k). - G. C. Greubel, May 04 2023

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 1-3 of 3 results.

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