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 A187250 Triangle read by rows: T(n,k) is the number of permutations of [n] having k cycles with at least 3 alternating runs (it is assumed that the smallest element of a cycle is in the first position), 0<=k<=floor(n/4). 2
 1, 1, 2, 6, 22, 2, 94, 26, 460, 260, 2532, 2508, 15420, 24760, 140, 102620, 254968, 5292, 739512, 2760432, 128856, 5729192, 31547344, 2640264, 47429896, 381339368, 50186136, 46200, 417429800, 4879612808, 926494712, 3483480, 3888426512, 66107044176, 17025751600, 157068912 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS Number of entries in row n is 1+floor(n/4). Sum of entries in row n is n!. T(n,0)=A187251(n). Sum(k*T(n,k), k>=0) = A187252(n). LINKS FORMULA E.g.f.: G(t,z) = exp[(1/4)(1-t)(2z-1+exp(2z))]/(1-z)^t. The 4-variate g.f. H(u,v,w,z) (exponential with respect z), where u marks number of cycles with 1 alternating run, v marks number of cycles with 2 alternating runs, w marks the number of all cycles, and z marks the size of the permutation, is given by H(u,v,w,z) = exp[(1/4)w((v-1)(exp(2z)+2z)+4(u-v)exp(z)+1-4u+3v)]/(1-z)^w. We have G(t,z) = H(1/t,1/t,t,z). EXAMPLE T(4,1)=2 because we have (1324) and (1423). Triangle starts: 1; 1; 2; 6; 22,2; 94,26; 460,260; MAPLE G := exp((1/4*(1-t))*(2*z-1+exp(2*z)))/(1-z)^t: Gser := simplify(series(G, z = 0, 17)): for n from 0 to 14 do P[n] := sort(factorial(n)*coeff(Gser, z, n)) end do: for n from 0 to 14 do seq(coeff(P[n], t, k), k = 0 .. floor((1/4)*n)) end do; # yields sequence in triangular form CROSSREFS Cf. A187244, A187247, A187251, A187252. Sequence in context: A182544 A216120 A216964 * A129534 A216719 A085286 Adjacent sequences:  A187247 A187248 A187249 * A187251 A187252 A187253 KEYWORD nonn,tabf AUTHOR Emeric Deutsch, Mar 08 2011 STATUS approved

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Last modified December 1 09:18 EST 2021. Contains 349426 sequences. (Running on oeis4.)