A105479 a(n) = C(n,2)*Bell(n-2) (cf. A000217, A000110).

0, 0, 1, 3, 12, 50, 225, 1092, 5684, 31572, 186300, 1163085, 7654350, 52928460, 383437327, 2902665885, 22907918640, 188082362120, 1603461748491, 14169892736484, 129594593170210, 1224875863061970, 11948280552370932, 120142063487658003, 1243853543811461148

Offset: 0

Author

Augustine O. Munagi, Apr 10 2005

Keywords

• easy
• nonn

Comments

Number of blocks of size 2 in all set partitions of {1,2,...,n}. Example: a(3)=3 because the set partitions of {1,2,3} are 1|2|3, 1|23, 12|3, 13|2 and 123, containing exactly 3 blocks of size 2. a(n) = Sum_{k>=0} k*A124498(n-1,k). - Emeric Deutsch, Nov 06 2006

Number of partitions of {1...n} containing 2 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. E.g. a(4) = 3 because the partitions of {1,2,3,4} with 2 pairs of consecutive integers are 123/4,12/34,1/234. - Augustine O. Munagi, Apr 10 2005

a(n) is the total sum of singletons in all set partitions of [n-1]. a(4) = 12 = 0+1+2+3+6: 123, 1|23, 13|2, 12|3, 1|2|3. - Alois P. Heinz, Mar 06 2024

Links

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

Crossrefs

Cf. A105480, A105489, A105484, A124498.

Column k=2 of A193297.

Programs

Maple

[seq(binomial(n,2)*combinat[bell](n-2),n=0..50)];

Mathematica

Join[{0,0},Table[Binomial[n,2]BellB[n-2],{n,2,30}]] (* Harvey P. Dale, May 06 2014 *)

Python

from itertools import count, accumulate, islice
def A105479_gen(): # generator of terms
    yield from (0,0,1)
    blist, b, c = (1,), 1, 1
    for n in count(2):
        c += n
        blist = list(accumulate(blist, initial=(b:=blist[-1])))
        yield b*c
A105479_list = list(islice(A105479_gen(),20)) # Chai Wah Wu, Jun 22 2022

Formula

a(n) = binomial(n-1, 2)*Bell(n-3), the case r = 2 of the general case of r pairs: c(n, r) = binomial(n-1, r)*Bell(n-r-1).

E.g.f.: z^2/2 * e^(e^z-1) - Frank Ruskey, Dec 26 2006

G.f.: exp(-1)*Sum_{n>=0} (x^2/(n!*(1-n*x)^3)). - 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>=2, a(n)=(-1)^(n-2)coeff(charpoly(A,x),x^2). - Milan Janjic, Jul 08 2010

G.f.: x^2/exp(1)*G(0), where G(k) = 1 + (2*k*x-1)^3/((2*k+1)*(2*k*x+x-1)^3 - (2*k+1)*(2*k*x+x-1)^6/((2*k*x+x-1)^3 + 2*(k+1)*(2*k*x+2*x-1)^3/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jun 14 2013

Extensions

Edited by N. J. A. Sloane, Jan 01 2007