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 A180185 Triangle read by rows: T(n,k) is the number of permutations of [n] having no 3-sequences and having k successions (0<=k<=floor(n/2)); a succession of a permutation p is a position i such that p(i +1) - p(i) = 1. 1
 1, 1, 1, 1, 3, 2, 11, 9, 1, 53, 44, 9, 309, 265, 66, 3, 2119, 1854, 530, 44, 16687, 14833, 4635, 530, 11, 148329, 133496, 44499, 6180, 265, 1468457, 1334961, 467236, 74165, 4635, 53, 16019531, 14684570, 5339844, 934472, 74165, 1854, 190899411 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,5 COMMENTS Row n has 1+floor(n/2) entries. Sum of entries in row n is A002628(n). LINKS FORMULA T(n,k) = binom(n-k,k)*[d(n-k)+d(n-k-1)], where d(j) = A000166(j) are the derangement numbers. T(n,0) = d(n)+d(n-1) = A000255(n-1). T(n,1) = d(n). Sum(k*T(n,k), k>=0) = A002629(n+1). EXAMPLE T(6,3)=3 because we have 125634, 341256, and 563412. Triangle starts: 1; 1; 1,1; 3,2; 11,9,1; 53,44,9; 309,265,66,3; 2119,1854,530,44; MAPLE d[0] := 1: for n to 51 do d[n] := n*d[n-1]+(-1)^n end do: a := proc (n, k) if n = 0 and k = 0 then 1 elif k <= (1/2)*n then binomial(n-k, k)*d[n+1-k]/(n-k) else 0 end if end proc: for n from 0 to 12 do seq(a(n, k), k = 0 .. (1/2)*n) end do; # yields sequence in triangular form PROG (PARI) d(n) = if(n<2, !n , round(n!/exp(1))); for(n=0, 20, for(k=0, (n\2), print1(binomial(n - k, k)*(d(n - k) + d(n - k - 1)), ", "); ); print(); ) \\ Indranil Ghosh, Apr 12 2017 CROSSREFS Cf. A000166, A002628, A002629, A000255 Sequence in context: A276589 A275950 A276587 * A072634 A086194 A258386 Adjacent sequences:  A180182 A180183 A180184 * A180186 A180187 A180188 KEYWORD nonn,tabf AUTHOR Emeric Deutsch, Sep 06 2010 STATUS approved

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Last modified October 15 03:16 EDT 2019. Contains 328025 sequences. (Running on oeis4.)