



0, 0, 1, 3, 12, 50, 225, 1092, 5684, 31572, 186300, 1163085, 7654350, 52928460, 383437327, 2902665885, 22907918640, 188082362120, 1603461748491, 14169892736484, 129594593170210, 1224875863061970, 11948280552370932
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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 123, 123, 123, 132 and 123, containing exactly 3 blocks of size 2. a(n)=Sum(k*A124498(n1,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


REFERENCES

A. O. Munagi, Set Partitions with Successions and Separations, Int. J. Math and Math. Sc. 2005, no. 3 (2005), 451463.


LINKS

Table of n, a(n) for n=0..22.
A. O. Munagi, Set Partitions with Successions and Separations,IJMMS 2005:3 (2005),451463.


FORMULA

a(n) = binomial(n1, 2)*Bell(n3), the case r = 2 of the general case of r pairs: c(n, r) = binomial(n1, r)B(nr1).
E.g.f.: z^2/2 * e^(e^z1)  Frank Ruskey, Dec 26 2006
G.f.: exp(1)*Sum(x^2/(n!*(1n*x)^3),n=0..infinity).  Vladeta Jovovic, Feb 05 2008
Let A be the upper Hessenberg matrix of order n defined by: A[i,i1]=1, A[i,j]=binomial(j1,i1), (i<=j), and A[i,j]=0 otherwise. Then, for n>=2, a(n)=(1)^(n2)coeff(charpoly(A,x),x^2). [From Milan Janjic, Jul 08 2010]
G.f.: x^2/exp(1)*G(0), where G(k)= 1 + (2*k*x1)^3/((2*k+1)*(2*k*x+x1)^3  (2*k+1)*(2*k*x+x1)^6/((2*k*x+x1)^3 + 2*(k+1)*(2*k*x+2*x1)^3/G(k+1))); (continued fraction).  Sergei N. Gladkovskii, Jun 14 2013


MAPLE

[seq(binomial(n, 2)*combinat[bell](n2), n=0..50)];


MATHEMATICA

Join[{0, 0}, Table[Binomial[n, 2]BellB[n2], {n, 2, 30}]] (* Harvey P. Dale, May 06 2014 *)


CROSSREFS

Cf. A105480, A105489, A105484, A124498.
Column k=2 of A193297.
Sequence in context: A151178 A151179 A191242 * A151180 A268650 A151181
Adjacent sequences: A105476 A105477 A105478 * A105480 A105481 A105482


KEYWORD

easy,nonn


AUTHOR

Augustine O. Munagi, Apr 10 2005


EXTENSIONS

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


STATUS

approved



