A180195 a(n)=(-1)^n*Sum((-1)^j*b(j), j=1..n), where b(n)=(n-1)!*(n^2 - n + 1) = A001564(n-1) (n>=1).

1, 2, 12, 66, 438, 3282, 27678, 259602, 2683758, 30338322, 372458478, 4936475922, 70266775278, 1069278031122, 17325341412078, 297824181275922, 5414097458148078, 103781942967323922, 2092232238097380078

Offset

1,2

Comments

a(n) is the number of blocks of odd size in all permutations of [n].

a(n) is the number of blocks of even size in all permutations of [n+1].

A block of a permutation is a maximal sequence of consecutive integers which appear in consecutive positions. Example: a(2)=2 because in 12 and (2)(1) we have a total of 2 blocks of odd size (shown between parentheses). Also, in 123, 132, 213, (23)1, 3(12), and 321 we have a total of 2 blocks of even size (shown between parentheses).

a(n)=Sum(k*A180193(n,k), k>=0).

a(n)=Sum(k*A180194(n+1,k), k>=0).

References

A. N. Myers, Counting permutations by their rigid patterns, J. Combin. Theory, A 99 (2002), 345-357.

Links

Table of n, a(n) for n=1..19.

Formula

Conjecture D-finite with recurrence a(n) +(-n-1)*a(n-1) -4*a(n-2) +(n-2)*a(n-3)=0. - R. J. Mathar, Jul 24 2022

Maple

b := proc (n) options operator, arrow: factorial(n-1)*(n^2-n+1) end proc: a := proc (n) options operator, arrow: (-1)^n*(sum((-1)^j*b(j), j = 1 .. n)) end proc; seq(a(n), n = 1 .. 20);

Crossrefs

Cf. A001564, A180193, A180194.

Sequence in context: A323925 A039633 A265451 * A212389 A241074 A020062

Adjacent sequences: A180192 A180193 A180194 * A180196 A180197 A180198

Keyword

nonn

Author

Emeric Deutsch, Sep 09 2010

Status

approved