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A105479 a(n) = C(n,2)*Bell(n-2) (cf. A000217, A000110). 12
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,4
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*A124498(n-1,k), k>=0}. - 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.
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
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 *)
PROG
(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
CROSSREFS
Column k=2 of A193297.
Sequence in context: A151178 A151179 A191242 * A151180 A268650 A151181
KEYWORD
easy,nonn,changed
AUTHOR
Augustine O. Munagi, Apr 10 2005
EXTENSIONS
Edited by N. J. A. Sloane, Jan 01 2007
STATUS
approved

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Last modified March 19 01:57 EDT 2024. Contains 370952 sequences. (Running on oeis4.)