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 1-5 of 5 results.

A246458 Catalan number analogs for A048804, the generalized binomial coefficients for the radical sequence (A007947).

Original entry on oeis.org

1, 1, 1, 5, 7, 7, 11, 143, 715, 2431, 4199, 29393, 52003, 37145, 7429, 215441, 392863, 4321493, 7960645, 58908773, 109402007, 407771117, 762354697, 3811773485, 35830670759, 19293438101, 327988447717, 2483341104143, 4709784852685, 17897182440203, 34062379482967
Offset: 0

Views

Author

Tom Edgar, Aug 26 2014

Keywords

Comments

One definition of the Catalan numbers is binomial(2*n,n) / (n+1); the current sequence models this definition using the generalized binomial coefficients arising from the radical sequence (A007947).

Examples

			A048804(10,5) = 42 and A007947(6) = 6, so a(5)=42/6=7.

Links

Crossrefs

Cf. A007947, A048804, A048803, A245798, A000108.

Programs

  • Sage
    [(1/(prod(x for x in prime_divisors(n+1))))*prod(prod(x for x in prime_divisors(i)) for i in [1..2*n])/prod(prod(x for x in prime_divisors(i)) for i in [1..n])^2 for n in [0..100]]

Formula

a(n) = A048804(2n,n) / A007947(n+1).

A048805 Matrix inverse of A048804.

Original entry on oeis.org

1, -1, 1, 1, -2, 1, -1, 3, -3, 1, 0, -2, 3, -2, 1, 4, 0, -5, 5, -5, 1, -24, 24, 0, -10, 15, -6, 1, 104, -168, 84, 0, -35, 21, -7, 1, -88, 208, -168, 56, 0, -14, 7, -2, 1, 42, -264, 312, -168, 84, 0, -7, 3, -3, 1, 536, 420, -1320, 1040, -840, 168, 0, -10, 15, -10, 1, -7536
Offset: 0

Views

Author

Christian G. Bower, Apr 15 1999

Keywords

Examples

			  1
 -1    1
  1   -2    1
 -1    3   -3    1
  0   -2    3   -2   1
  4    0   -5    5  -5   1
-24   24    0  -10  15  -6  1
104 -168   84    0 -35  21 -7  1
-88  208 -168   56   0 -14  7 -2  1
 42 -264  312 -168  84   0 -7  3 -3 1

A048807 Matrix square of A048804.

Original entry on oeis.org

1, 2, 1, 4, 4, 1, 8, 12, 6, 1, 9, 16, 12, 4, 1, 22, 45, 40, 20, 10, 1, 54, 132, 135, 80, 60, 12, 1, 128, 378, 462, 315, 280, 84, 14, 1, 83, 256, 378, 308, 315, 112, 28, 4, 1, 70, 249, 384, 378, 462, 189, 56, 12, 6, 1, 184, 700, 1245, 1280, 1890, 924, 315, 80, 60
Offset: 0

Views

Author

Christian G. Bower, Apr 15 1999

Keywords

Examples

			1; 2,1; 4,4,1; 8,12,6,1; ...

A246465 Triangle read by rows: T(n,k) = A085056(n)/(A085056(k) * A085056(n-k)).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 4, 4, 4, 2, 4, 4, 4, 1, 1, 3, 12, 12, 6, 6, 12, 12, 3, 1, 1, 1, 3, 12, 6, 6, 6, 12, 3, 1, 1, 1, 1, 1, 3, 6, 6, 6, 6, 3, 1, 1, 1, 1, 2, 2, 2, 3, 12, 12
Offset: 0

Views

Author

Tom Edgar, Aug 27 2014

Keywords

Comments

We assume that A085056(0)=1 since it would be the empty product.
These are the generalized binomial coefficients associated with the sequence A003557.

Examples

			The first five terms in A003557 are: 1, 1, 1, 2, 1 and so T(4,2) = 2*1*1*1/((1*1)*(1*1))=2 and T(5,4) = 1*2*1*1*1/((2*1*1*1)*(1))=1.
The triangle begins:
1,
1, 1,
1, 1, 1,
1, 1, 1, 1,
1, 2, 2, 2, 1,
1, 1, 2, 2, 1, 1,
1, 1, 1, 2, 1, 1, 1,
1, 1, 1, 1, 1, 1, 1, 1,
1, 4, 4, 4, 2, 4, 4, 4, 1,
1, 3, 12, 12, 6, 6, 12, 12, 3, 1.

Links

Crossrefs

Cf. A003557, A085056, A048804.

Programs

  • Sage
    q=100 #change q for more rows
P=[0]+[n/prod([x for x in prime_divisors(n)]) for n in [1..q]]
[[prod(P[1:n+1])/(prod(P[1:k+1])*prod(P[1:(n-k)+1])) for k in [0..n]] for n in [0..len(P)-1]] # generates the triangle up to q rows.

Formula

T(n,k) = A085056(n)/(A085056(k) * A085056(n-k)).
T(n,k) = prod_{i=1..n} A003557(i)/(prod_{i=1..k} A003557(i)*prod_{i=1..n-k} A003557(i)).
T(n,k) = A003557(n)/n*(k/A003557(k)*T(n-1,k-1)+(n-k)/A003557(n-k)*T(n-1,k)).

A239702 Triangle read by rows: T(n,k) = A239682(n)/(A239682(k)* A239682(n-k)).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 2, 1, 1, 1, 4, 4, 4, 4, 1, 1, 2, 8, 4, 8, 2, 1, 1, 6, 12, 24, 24, 12, 6, 1, 1, 1, 6, 6, 24, 6, 6, 1, 1, 1, 2, 2, 6, 12, 12, 6, 2, 2, 1, 1, 4, 8, 4, 24, 12, 24, 4, 8, 4, 1, 1, 10, 40, 40, 40, 60, 60, 40, 40, 40, 10, 1, 1, 2
Offset: 0

Views

Author

Tom Edgar, Mar 24 2014

Keywords

Comments

A239682(0) = 1 since it is the empty product.
These are the generalized binomial coefficients associated with the sequence A173557.

Examples

			The first six terms A173557 are 1,1,2,1,4,2 and so T(4,2) = 1*2*1*1/((1*1)*(1*1))=2 and T(6,3) = 2*4*1*2*1*1/((2*1*1)*(2*1*1))=4.
The triangle begins
1
1 1
1 1  1
1 2  2  1
1 1  2  1  1
1 4  4  4  4  1
1 2  8  4  8  2 1
1 6 12 24 24 12 6 1

Links

Crossrefs

Cf. A239682, A173557, A048804, A238453.

Programs

  • Sage
    q=100 #change q for more rows
P=[0]+[prod([(x-1) for x in prime_divisors(n)]) for n in [1..q]]
[[prod(P[1:n+1])/(prod(P[1:k+1])*prod(P[1:(n-k)+1])) for k in [0..n]] for n in [0..len(P)-1]] # generates the triangle up to q rows.

Formula

T(n,k) = A239682(n)/(A239682(k)* A239682(n-k)).
T(n,k) = prod_{i=1..n} A173557(i)/(prod_{i=1..k} A173557(i)*prod_{i=1..n-k} A173557(i)).
T(n,k) = A173557(n)/n*(k/A173557(k)*T(n-1,k-1)+(n-k)/A173557(n-k)*T(n-1,k)).
Showing 1-5 of 5 results.

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