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.

A234616 Numbers of undirected cycles in the complete tripartite graph K_{n,n,n}.

Original entry on oeis.org

1, 63, 6705, 1960804, 1271288295, 1541975757831, 3135880743480163, 9904953891455450640, 45915662047529291081589, 299038026557168514632822455, 2642895689915240835222121682301, 30814273315381549790551229559722628
Offset: 1

Views

Author

Eric W. Weisstein, Dec 28 2013

Keywords

Links

Crossrefs

Cf. A234365, A234633, A070968.
Cf. A296546 (cycle polynomial coefficients of K_n,n,n).

Programs

  • Mathematica
    Table[(Sum[Binomial[n, k] Binomial[n, i + p] Binomial[n, j + p] Binomial[k, i] Binomial[k - i, j] (k - 1)! (i + p)! (j + p)! 2^(k - i - j) Binomial[p + i + j - 1, k - 1], {k, n}, {i, 0, k}, {j, 0, k - i}, {p, k - i - j, n}] + Sum[Binomial[n, k]^2 k! (k - 1)!, {k, 2, n}])/2 - n^2, {n, 10}] (* Eric W. Weisstein, May 26 2017 *)
Table[(n^2 (HypergeometricPFQ[{1, 1, 1 - n, 1 - n}, {2}, 1] - 3) + Sum[2^(k - i - j) Binomial[k, i] Binomial[k - i, j] Binomial[n, k] Binomial[n, i + p] Binomial[n, j + p] Binomial[i + j + p - 1, k - 1] (k - 1)! (i + p)! (j + p)!, {k, n}, {i, 0, k}, {j, 0, k - i}, {p, k - i - j, n}])/2, {n, 10}] (* Eric W. Weisstein, May 25 2023 *)
  • PARI
    c(n,k,i,j,p) = {binomial(n,k)*binomial(n,i+p)*binomial(n,j+p)*binomial(k,i)*binomial(k-i,j)*(k-1)!*(i+p)!*(j+p)!*2^(k-i-j)*binomial(p+i+j-1,k-1)}
a(n)={(sum(k=1,n,sum(i=0,k,sum(j=0,k-i,sum(p=k-i-j,n, c(n,k,i,j,p) )))) + sum(k=2,n,binomial(n,k)^2*k!*(k-1)!))/2 - n^2} \\ Andrew Howroyd, May 25 2017
  • Python
    from sympy import binomial, factorial
def c(n, k, i, j, p): return binomial(n, k)*binomial(n, i + p)*binomial(n, j + p)*binomial(k, i)*binomial(k - i, j)*factorial(k - 1)*factorial(i + p)*factorial(j + p)*2**(k - i - j)*binomial(p + i + j - 1, k - 1)
def a(n): return (sum([sum([sum([sum([c(n, k, i, j, p) for p in range(k - i - j, n + 1)]) for j in range(k - i + 1)]) for i in range(k + 1)]) for k in range(1, n + 1)]) + sum(binomial(n, k)**2*factorial(k)*factorial(k - 1) for k in range(2, n + 1)))/2 - n**2
print([a(k) for k in range(1, 13)]) # Indranil Ghosh, Aug 14 2017, after PARI

Formula

Row sums of A296546.
a(n) ~ sqrt(3*Pi) * 2^(3*n - 1/2) * n^(3*n - 1/2) / exp(3*n - 3/2). - Vaclav Kotesovec, Feb 17 2024

Extensions

a(7)-a(12) from Andrew Howroyd, May 25 2017

A234633 Numbers of directed Hamiltonian paths in the complete tripartite graph K_{n,n,n}.

Original entry on oeis.org

6, 240, 37584, 15095808, 12420864000, 18233911296000, 43492335022080000, 157551157218115584000, 823642573772373884928000, 5970637844437187690496000000, 58120324656942369834270720000000, 739968068159742816891489484800000000
Offset: 1

Views

Author

Eric W. Weisstein, Dec 28 2013

Keywords

Links

Crossrefs

Cf. A234365, A234616, A110706.

Programs

  • Mathematica
    Table[2 n!^3 (Binomial[2 n + 1, n + 1] HypergeometricPFQ[{1 - n, 1 - n, 1/2 - n/2, -(n/2)}, {1, -(1/2) - n, -n}, 1] + (n - 1) Binomial[2 n, n + 1] HypergeometricPFQ[{1 - n, 2 - n, 1/2 - n/2, 1 - n/2}, {2, 1/2 - n, -n}, 1]), {n, 10}] (* Eric W. Weisstein, May 26 2017 *)

Formula

a(n) = n!^3 * A110706(n). - Andrew Howroyd, May 24 2017

Extensions

a(7)-a(12) from Andrew Howroyd, May 24 2017

A377586 Numbers of directed Hamiltonian paths in the complete 4-partite graph K_{n,n,n,n}.

Original entry on oeis.org

24, 13824, 53529984, 751480602624, 27917203599360000, 2267561150913576960000, 354252505303682314076160000, 97087054992658680467800719360000, 43551509948777170973522371396239360000, 30293653795894300342540281328749772800000000
Offset: 1

Views

Author

Zlatko Damijanic, Nov 02 2024

Keywords

Links

Crossrefs

Cf. A190918, A234365, A322013, A322093.

Programs

  • Mathematica
    Table[n!^4 * SeriesCoefficient[1/(1 - Sum[x[i]/(1 + x[i]), {i, 1, 4}]), Sequence @@ Table[{x[i], 0, n}, {i, 1, 4}]], {n, 1, 10}]
  • Python
    from math import factorial as fact, comb
from itertools import combinations_with_replacement
def a(n):
    #  Using modified formula for counting sequences found in Eifler et al.
    result = 0
    fn = fact(n)
    for i, j, k in combinations_with_replacement(range(1, n+1), 3):
        patterns = [(3,0,0)] if i == j == k else \
          [(2,0,1)] if i == j != k else \
          [(1,2,0)] if i != j == k else [(1,1,1)]
        for a, b, c in patterns:
            s = a*i + b*j + c*k
            num = fact(3)
            den = fact(a) * fact(b) * fact(c)
            if a:
                for _ in range(a): num, den = num * comb(n-1, i-1), den * fact(i)
            if b:
                for _ in range(b): num, den = num * comb(n-1, j-1), den * fact(j)
            if c:
                for _ in range(c): num, den = num * comb(n-1, k-1), den * fact(k)
            num *= comb(s + 1, n) * fact(s)
            result += (1 if (3*n - s) % 2 == 0 else -1) * (num // den)
    for _ in range(4): result *= fn
    return result
print([a(n) for n in range(1,11)]) # Zlatko Damijanic, Nov 18 2024

Formula

a(n) = 24 * n!^4 * A190918(n).
a(n) = n!^4 * A322093(n,4).

A307924 Number of (undirected) Hamiltonian cycles in the complete tripartite graph K_{n,n,n}.

Original entry on oeis.org

1, 16, 1584, 463104, 299289600, 361552896000, 732443959296000, 2305449150971904000, 10654390419268829184000, 69202145783548005580800000, 610152377325314475294720000000, 7098963015274747190071787520000000, 106459726394067298796772293345280000000, 2017443502989317777418537171765166080000000
Offset: 1

Views

Author

Eric W. Weisstein, May 06 2019

Keywords

Links

Crossrefs

Cf. A234365.

Programs

  • Mathematica
    Table[2^(n - 1) (n - 1)! (n!)^2 HypergeometricPFQ[{1/2 - n/2, -n/2, n}, {1, 1}, 1], {n, 20}] (* Eric W. Weisstein, Feb 19 2025 *)

Formula

a(n) = A234365(n)/2.

A378241 Numbers of directed Hamiltonian cycles in the complete 4-partite graph K_{n,n,n,n}.

Original entry on oeis.org

6, 1488, 3667680, 37744330752, 1106491456512000, 74213488705904640000, 9872975878366503813120000, 2355966665497190945783808000000, 935825492908108988335792827924480000, 584053924678169568704863421815848960000000
Offset: 1

Views

Author

Zlatko Damijanic, Nov 20 2024

Keywords

Links

Crossrefs

Cf. A209183, A234365, A369923, A377586.

Programs

  • Mathematica
    Table[Sum[Sum[Sum[Sum[ (2n - i - j - 1)! 2^(2i) 3^j (n!)^4/(j!) * (3n - 3i - 3j - 2d)!/((2i + j - n + d)! (n - j - d)! (2n - 3i - 2j - d)!) * (2n - 2i - 2j - 2e)!/(e! (d - e)! (2n - 2i - 2j - d - e)! (n - i - j - d + e)! ((n - i - j - e)!)^2), {e, Max[0, i + j - n + d], Min[d, 2n - 2i - 2j - d]}], {d, Max[0, n - j - 2i], Min[n - j, 2n - 3i - 2j]}], {i, 0, Floor[2(n - j)/3]}], {j, 0, n}], {n, 1, 10}]
Table[(n!)^4 Expand[Hypergeometric1F1[1 - n, 2, x]^4 x^3] /. x^p_ :> p!, {n, 10}] (* Eric W. Weisstein, Feb 20 2025 *)
  • Python
    from math import factorial as fact
def a(n):
   # Using formula found in Horak et al.
   return sum(sum(sum(sum(
       fact(2*n-i-j-1)*pow(2,2*i)*pow(3,j)*pow(fact(n),4)//fact(j) *
       fact(3*n-3*i-3*j-2*d)//(fact(2*i+j-n+d)*fact(n-j-d)*fact(2*n-3*i-2*j-d)) *
       fact(2*n-2*i-2*j-2*e)//(fact(e)*fact(d-e)*fact(2*n-2*i-2*j-d-e)*fact(n-i-j-d+e)*pow(fact(n-i-j-e),2))
       for e in range(max(0,i+j-n+d), min(d,2*n-2*i-2*j-d)+1))
       for d in range(max(0,n-j-2*i), min(n-j,2*n-3*i-2*j)+1))
       for i in range(int(2*(n-j)/3)+1))
       for j in range(n+1))
print([a(n) for n in range(1,11)])

Formula

a(n) = 3!*(n-1)!*(n!)^3*A369923(n,4). - Andrew Howroyd, Nov 20 2024
a(n) = 2*A381326(n). - Eric W. Weisstein, Feb 20 2025
Showing 1-5 of 5 results.

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