A105482 Number of partitions of {1...n} containing 5 pairs of consecutive integers, where each pair is counted within a block and a string of more than 2 consecutive integers are counted two at a time.

1, 6, 42, 280, 1890, 13104, 93786, 694584, 5328180, 42336294, 348272925, 2963993760, 26073738236, 236857536216, 2219777316216, 21441389281680, 213260412549303, 2182163481418536, 22951202450444191, 247914874683742728

Offset

6,2

Links

Table of n, a(n) for n=6..25.

A. O. Munagi, Set Partitions with Successions and Separations, IJMMS 2005:3 (2005), 451-463.

Formula

a(n) = binomial(n-1, 5)*Bell(n-6), the case r = 5 in the general case of r pairs: c(n, r) = binomial(n-1, r)*B(n-r-1).

Let A be the upper Hessenberg matrix of order n defined by: A[i,i-1]=-1, A[i,j]=binomial(j-1,i-1), (i<=j), and A[i,j]=0 otherwise. Then, for n>=5, a(n+1)=(-1)^(n-5)*coeff(charpoly(A,x),x^5). [Milan Janjic, Jul 08 2010]

E.g.f.: (1/5!) * Integral (x^5 * exp(exp(x) - 1)) dx. - Ilya Gutkovskiy, Jul 10 2020

Example

a(7) = 6 because the partitions of {1,2,3,4,5,6,7} with 5 pairs of consecutive integers are 123456/7,12345/67,1234/567,123/4567,12/34567,1/234567.

Maple

seq(binomial(n-1, 5)*combinat[bell](n-6), n=6..26);

Crossrefs

Cf. A105481, A105487, A105491.

Sequence in context: A062310 A377198 A229247 * A242158 A373715 A157335

Adjacent sequences: A105479 A105480 A105481 * A105483 A105484 A105485

Keyword

easy,nonn

Author

Augustine O. Munagi, Apr 10 2005

Status

approved