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.

A254706 a(n) = Catalan(2*n) mod prime(n).

Original entry on oeis.org

0, 2, 2, 2, 10, 12, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 32, 33, 122, 35, 125, 64, 70, 90, 36, 71, 50, 65, 159, 27, 79, 175, 155, 197, 164, 66, 95, 204, 156, 223, 191, 156, 140, 184, 231, 32, 208, 35, 224, 83, 193, 143, 234, 1, 273
Offset: 1

Views

Author

Vincenzo Librandi, Feb 06 2015

Keywords

Links

Crossrefs

Cf. A000040, A000108, A246714.

Programs

  • Magma
    [Catalan(2*n) mod NthPrime(n): n in [1..100]];
  • Mathematica
    Table[Mod[CatalanNumber[2 n], Prime[n]], {n, 70}]

Formula

a(n) = A000108(2n) mod A000040(n).

A246763 Catalan(n)^2 mod prime(n).

Original entry on oeis.org

1, 1, 0, 0, 4, 4, 16, 6, 12, 25, 7, 4, 31, 15, 8, 11, 7, 41, 23, 45, 69, 72, 29, 11, 2, 85, 4, 16, 73, 64, 2, 62, 69, 5, 29, 144, 16, 145, 157, 40, 9, 82, 75, 96, 88, 9, 100, 144, 36, 118, 8, 163, 212, 38, 9, 27, 185, 242, 203, 231, 11, 189, 250, 137, 116, 34, 91, 289, 10, 272
Offset: 1

Views

Author

Vincenzo Librandi, Sep 03 2014

Keywords

Links

Crossrefs

Cf. A000040, A000108, A246714.

Programs

  • Magma
    [Catalan(n)^2 mod NthPrime(n): n in [1..70]];
  • Maple
    seq(binomial(2*n,n)^2/(n+1)^2 mod ithprime(n), n=1..100); # Robert Israel, Sep 03 2014
  • Mathematica
    Table[Mod[CatalanNumber[n]^2, Prime[n]], {n, 70}]
  • Python
    from sympy import prime
from gmpy2 import divexact, t_mod
A246763, c = [1], 1
for n in range(2,10**2):
    c = divexact(c*(4*n-2),(n+1))
    A246763.append(t_mod(c**2,prime(n))) # Chai Wah Wu, Sep 04 2014

A254746 a(n) = Catalan(n^2) mod prime(n).

Original entry on oeis.org

1, 2, 2, 5, 8, 0, 0, 11, 0, 24, 0, 0, 29, 0, 0, 0, 0, 39, 58, 0, 21, 33, 9, 69, 9, 0, 16, 86, 0, 0, 0, 0, 0, 64, 139, 0, 0, 0, 75, 12, 4, 0, 0, 119, 195, 0, 193, 202, 0, 0, 55, 218, 0, 0, 0, 0, 84, 201, 0, 0, 203, 275, 0, 198, 159, 0, 0, 0, 0, 255, 13, 204, 0
Offset: 1

Views

Author

Vincenzo Librandi, Feb 07 2015

Keywords

Links

Crossrefs

Cf. A000040, A214441, A246714, A254706.

Programs

  • Magma
    [Catalan(n^2) mod NthPrime(n): n in [1..100]];
  • Mathematica
    Table[Mod[CatalanNumber[n^2], Prime[n]], {n, 80}]
  • Python
    from sympy import factorint, prime
A254746_list, c, s, s2 = [1], {}, 2, 4
for n in range(2,10**3+1):
    for p,e in factorint(4*n-2).items():
        if p in c:
            c[p] += e
        else:
            c[p] = e
    for p,e in factorint(n+1).items():
        if c[p] == e:
            del c[p]
        else:
            c[p] -= e
    if n == s2:
        d, ps = 1, prime(s)
        for p,e in c.items():
            d = (d*pow(p,e,ps)) % ps
        A254746_list.append(d)
        s2 += 2*s+1
        s += 1 # Chai Wah Wu, Feb 14 2015

Formula

a(n) = A000108(n^2) mod A000040(n).

A336257 a(n) = Catalan(n) mod (2*n+1).

Original entry on oeis.org

0, 1, 2, 5, 5, 9, 2, 9, 2, 17, 17, 21, 12, 22, 2, 29, 18, 30, 2, 30, 2, 41, 30, 45, 9, 21, 2, 54, 53, 57, 2, 28, 38, 65, 42, 69, 2, 64, 70, 77, 5, 81, 80, 33, 2, 14, 27, 45, 2, 36, 2, 101, 87, 105, 2, 78, 2, 34, 75, 6, 101, 45, 62, 125, 39, 129, 74, 60, 2, 137, 90
Offset: 0

Views

Author

Michel Marcus, Jul 15 2020

Keywords

Links

Crossrefs

Cf. A000108, A059288, A067770, A246714.
Cf. A104635, A104636.

Programs

  • Maple
    a:= n-> binomial(2*n, n)/(n+1) mod (2*n+1):
seq(a(n), n=0..80);  # Alois P. Heinz, Jul 16 2020
  • PARI
    C(n)=binomial(2*n, n)/(n+1);
a(n) = C(n) % (2*n+1);
  • Python
    A336257_list, c = [0,1], 1
for n in range(2,10001):
    c = c*(4*n-2)//(n+1)
    A336257_list.append(c % (2*n+1)) # Chai Wah Wu, Jul 16 2020

Formula

a(n) = 2 for n in A104636.
a(n) = 2*n-1 for n in A104635.
Showing 1-4 of 4 results.

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