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 A233440 Triangle read by rows: T(n, k) = n*binomial(n, k)*A000757(k), 0 <= k <= n. 13
 0, 1, 0, 2, 0, 0, 3, 0, 0, 3, 4, 0, 0, 16, 4, 5, 0, 0, 50, 25, 40, 6, 0, 0, 120, 90, 288, 216, 7, 0, 0, 245, 245, 1176, 1764, 1603, 8, 0, 0, 448, 560, 3584, 8064, 14656, 13000, 9, 0, 0, 756, 1134, 9072, 27216, 74196, 131625, 118872, 10, 0, 0, 1200, 2100, 20160, 75600, 274800, 731250, 1320800, 1202880 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,4 COMMENTS For n >= 0, 0 <= k <= n, T(n, k) is the number of permutations of n symbols that k-commute with an n-cycle (we say that two permutations f and g k-commute if H(fg, gf) = k, where H(, ) denotes the Hamming distance between permutations). Row sums give A000142. LINKS Luis Manuel Rivera Martínez, Rows n = 0..30 of triangle, flattened R. Moreno and L. M. Rivera, Blocks in cycles and k-commuting permutations, arXiv:1306.5708 [math.CO], 2013-2014. Luis Manuel Rivera, Integer sequences and k-commuting permutations, arXiv preprint arXiv:1406.3081 [math.CO], 2014-2015. FORMULA T(n,k) = n*C(n,k)*A000757(k), 0 <= k <= n. Bivariate e.g.f.: G(z, u) = z*exp(z*(1-u))*(u/(1-z*u)+(1-log(1-z*u))*(1-u)). T(n, 0)  = A001477(n), n>=0; T(n, 1)  = A000004(n), n>=1; T(n, 2)  = A000004(n), n>=2; T(n, 3)  = A004320(n-2), n>=3; T(n, 4)  = A027764(n-1), n>=4; T(n, 5)  = A027765(n-1)*A000757(5), n>=5; T(n, 6)  = A027766(n-1)*A000757(6), n>=6; T(n, 7)  = A027767(n-1)*A000757(7), n>=7; T(n, 8)  = A027768(n-1)*A000757(8), n>=8; T(n, 9)  = A027769(n-1)*A000757(9), n>=9; T(n, 10) = A027770(n-1)*A000757(10), n>=10; T(n, 11) = A027771(n-1)*A000757(11), n>=11; T(n, 12) = A027772(n-1)*A000757(12), n>=12; T(n, 13) = A027773(n-1)*A000757(13), n>=13; T(n, 14) = A027774(n-1)*A000757(14), n>=14; T(n, 15) = A027775(n-1)*A000757(15), n>=15; T(n, 16) = A027776(n-1)*A000757(16), n>=16. - Luis Manuel Rivera Martínez, Feb 08 2014 T(n, 0)+T(n, 3) = n*A050407(n+1), for n>=0. - Luis Manuel Rivera Martínez, Mar 06 2014 EXAMPLE For n = 4 and k = 4, T(4, 4) = 4 because all the permutations of 4 symbols that 4-commute with permutation (1, 2, 3, 4) are (1, 3), (2, 4), (1, 2)(3, 4) and (1, 4)(2, 3). MATHEMATICA T[n_, k_] := n Binomial[n, k] ((-1)^k+Sum[(-1)^j k!/(k-j)/j!, {j, 0, k-1}]); Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Nov 03 2018 *) CROSSREFS Cf. A007318, A000757. Sequence in context: A280164 A049597 A210951 * A280728 A175676 A035377 Adjacent sequences:  A233437 A233438 A233439 * A233441 A233442 A233443 KEYWORD nonn,tabl AUTHOR Luis Manuel Rivera Martínez, Dec 09 2013 STATUS approved

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Last modified October 15 10:46 EDT 2019. Contains 328026 sequences. (Running on oeis4.)