A105481 Number of partitions of {1...n} containing 4 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, 5, 30, 175, 1050, 6552, 42630, 289410, 2049300, 15120105, 116090975, 926248050, 7668746540, 65793760060, 584151925320, 5360347320420, 50776288702215, 495946245776940, 4989391837053085, 51648932225779735, 549620905409062872

Offset

5,2

References

A. O. Munagi, Set Partitions with Successions and Separations, Int. J. Math and Math. Sc. 2005, no. 3 (2005), 451-463.

Links

Table of n, a(n) for n=5..25.

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

Formula

a(n) = binomial(n-1, 4)*Bell(n-5), the case r = 4 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>=4, a(n+1)=(-1)^(n-4)*coeff(charpoly(A,x),x^4). [Milan Janjic, Jul 08 2010]

E.g.f.: (1/4!) * Integral (x^4 * exp(exp(x) - 1)) dx. - Ilya Gutkovskiy, Jul 10 2020

Example

a(6) = 5 because the partitions of {1,2,3,4,5,6} with 4 pairs of consecutive integers are 12345/6,1234/56,123/456,12/3456,1/23456.

Maple

seq(binomial(n-1, 4)*combinat[bell](n-5), n=5..25);

Crossrefs

Cf. A105480, A105482, A105486, A105491, A105494.

Sequence in context: A276598 A057088 A156195 * A242157 A094167 A051738

Adjacent sequences: A105478 A105479 A105480 * A105482 A105483 A105484

Keyword

easy,nonn

Author

Augustine O. Munagi, Apr 10 2005

Status

approved