A105480 Number of partitions of {1...n} containing 3 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, 4, 20, 100, 525, 2912, 17052, 105240, 683100, 4652340, 33168850, 246999480, 1917186635, 15480884720, 129811538960, 1128494172720, 10155257740443, 94465951576560, 907162152191470, 8982422995787780, 91603484234843812

Offset

4,2

Links

Table of n, a(n) for n=4..24.

A. O. Munagi, Set partitions with successions and separations, Int. J. Math. Math. Sci. (IJMMS) vol 2005 no 3 (2005) pp 451-463.

Formula

a(n) = binomial(n-1, 3)*Bell(n-4), the case r = 3 in the general case of r pairs: c(n, r) = binomial(n-1, r)*B(n-r-1).

O.g.f. for c(n,r) is exp(-1)*Sum(x^(r+1)/(n!*(1-n*x)^(r+1)),n=0..infinity). - Vladeta Jovovic, Feb 05 2008

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>=3, a(n+1)=(-1)^(n-3)*coeff(charpoly(A,x),x^3). [Milan Janjic, Jul 08 2010]

E.g.f.: (1/3!) * Integral (x^3 * exp(exp(x) - 1)) dx. - Ilya Gutkovskiy, Jul 10 2020

Example

a(5) = 4 because the partitions of {1,2,3,4,5} with 3 pairs of consecutive integers are 1234/5,123/45,12/345,1/2345.

Maple

seq(binomial(n-1, 3)*combinat[bell](n-4), n=4..25);

Crossrefs

Cf. A105479, A105481, A105485, A105490.

Sequence in context: A005054 A370536 A216099 * A242156 A186369 A093440

Adjacent sequences: A105477 A105478 A105479 * A105481 A105482 A105483

Keyword

easy,nonn

Author

Augustine O. Munagi, Apr 10 2005

Status

approved