Zlatko Damijanic has authored 3 sequences.
A378377
Triangle read by rows: T(n,k) is the number of non-descending sequences with length k such that the maximum of the length and the last number is n.
Original entry on oeis.org
1, 1, 3, 1, 3, 10, 1, 4, 10, 35, 1, 5, 15, 35, 126, 1, 6, 21, 56, 126, 462, 1, 7, 28, 84, 210, 462, 1716, 1, 8, 36, 120, 330, 792, 1716, 6435, 1, 9, 45, 165, 495, 1287, 3003, 6435, 24310, 1, 10, 55, 220, 715, 2002, 5005, 11440, 24310, 92378
Offset: 1
Triangle begins:
1
1 3
1 3 10
1 4 10 35
1 5 15 35 126
1 6 21 56 126 462
1 7 28 84 210 462 1716
...
For T(3,1) solution is |{(3)}| = 1.
For T(3,2) solution is |{(1,3), (2,3), (3,3)}| = 3.
For T(3,3) solution is |{(1,1,1), (1,1,2), (1,1,3), (1,2,2), (1,2,3), (1,3,3), (2,2,2), (2,2,3), (2,3,3), (3,3,3)}| = 10.
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T[n_, k_] := Which[
k == 1, 1,
k == n, Binomial[2n-1, n],
k == n-1, T[n-1, n-1],
1 < k < n-1, T[n-1, k] + T[n, k-1]
];
Table[T[n, k], {n, 1, 10}, {k, 1, n}] // Flatten
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T(n,k)={if(kAndrew Howroyd, Nov 24 2024
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
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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 *)
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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)])
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
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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}]
-
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
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