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
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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 *)
A296546
Triangle read by rows T(n,k): number of undirected cycles of length k in the complete tripartite graph K_{n,n,n} (n = 1...; k = 3..3n).
Original entry on oeis.org
1, 8, 15, 24, 16, 27, 108, 324, 774, 1620, 2268, 1584, 64, 396, 1728, 7200, 27648, 87480, 232704, 476928, 663552, 463104, 125, 1050, 6000, 35800, 198000, 977400, 4392000, 17068320, 56376000, 151632000, 311040000, 430272000, 299289600
Offset: 1
Written as cycle polynomials:
x^3
8 x^3 + 15 x^4 + 24 x^5 + 16 x^6
27 x^3 + 108 x^4 + 324 x^5 + 774 x^6 + 1620 x^7 + 2268 x^8 + 1584 x^9
64 x^3 + 396 x^4 + 1728 x^5 + 7200 x^6 + 27648 x^7 + 87480 x^8 + 232704 x^9 + 476928 x^10 + 663552 x^11 + 463104 x^12
giving the array
1
8, 15, 24, 16
27, 108, 324, 774, 1620, 2268, 1584
64, 396, 1728, 7200, 27648, 87480, 232704, 476928, 663552, 463104
Cf.
A234616 (number of undirected cycles in K_{n,n,n}).
Cf.
A144151 (cycle polynomial coefficients of complete graph K_n).
Cf.
A291909 (cycle polynomial coefficients of complete bipartite graph K_{n,n}).
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Table[Tally[Length /@ FindCycle[CompleteGraph[{n, n, n}], Infinity, All]][[All, 2]], {n, 4}] // Flatten
A290324
Number of (undirected) paths in the complete tripartite graph K_{n,n,n}.
Original entry on oeis.org
6, 396, 67554, 28336848, 23986682550, 35931274250076, 87017657752978386, 318992998488391738944, 1683675635040443830593798, 12301777316059025283613106700, 120545445185882926234784081616546, 1543421856441404929844846263740167376
Offset: 1
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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!*(i + p)!*(j + p)!*2^(k - i - j)*Binomial[p + i + j - 1, k - 1](1 + n - k); a[n_]:=3*Sum[Sum[Sum[Sum[c[n, k, i, j, p], {p, k - i - j, n}], {j, 0, k - i}], {i, 0, k}], {k, n}]/2; Table[a[n], {n, 12}] (* Indranil Ghosh, Aug 14 2017, after PARI *)
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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!*(i+p)!*(j+p)!*2^(k-i-j)*binomial(p+i+j-1, k-1)*(1+n-k)}
a(n)={3*(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) )))))/2} \\ Andrew Howroyd, Aug 13 2017
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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)*factorial(i + p)*factorial(j + p)*2**(k - i - j)*binomial(p + i + j - 1, k - 1)*(1 + n - k)
def a(n): return 3*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)])/2
print([a(n) for n in range(1, 13)]) # Indranil Ghosh, Aug 14 2017, after PARI
A297662
Number of chordless cycles in the complete tripartite graph K_{n,n,n}.
Original entry on oeis.org
0, 3, 27, 108, 300, 675, 1323, 2352, 3888, 6075, 9075, 13068, 18252, 24843, 33075, 43200, 55488, 70227, 87723, 108300, 132300, 160083, 192027, 228528, 270000, 316875, 369603, 428652, 494508, 567675, 648675, 738048, 836352, 944163, 1062075, 1190700
Offset: 1
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Table[3 Binomial[n, 2]^2, {n, 20}]
3 Binomial[Range[20], 2]^2
LinearRecurrence[{5, -10, 10, -5, 1}, {0, 3, 27, 108, 300}, 20]
SeriesCoefficient[Series[-3 x (1 + 4 x + x^2)/(-1 + x)^5, {x, 0, 20}], x]
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a(n) = 3*n^2*(n-1)^2/4; \\ Andrew Howroyd, Jan 03 2018
Showing 1-4 of 4 results.
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