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%I #35 Mar 06 2024 10:07:45
%S 0,0,1,3,12,50,225,1092,5684,31572,186300,1163085,7654350,52928460,
%T 383437327,2902665885,22907918640,188082362120,1603461748491,
%U 14169892736484,129594593170210,1224875863061970,11948280552370932,120142063487658003,1243853543811461148
%N a(n) = C(n,2)*Bell(n-2) (cf. A000217, A000110).
%C 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
%C 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
%C 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
%H Augustine O. Munagi, <a href="http://www.emis.de/journals/HOA/IJMMS/2005/3451.pdf">Set Partitions with Successions and Separations</a>, Int. J. Math and Math. Sc. 2005, no. 3 (2005), 451-463.
%F 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).
%F E.g.f.: z^2/2 * e^(e^z-1) - _Frank Ruskey_, Dec 26 2006
%F G.f.: exp(-1)*Sum_{n>=0} (x^2/(n!*(1-n*x)^3)). - _Vladeta Jovovic_, Feb 05 2008
%F 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]
%F 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
%p [seq(binomial(n,2)*combinat[bell](n-2),n=0..50)];
%t Join[{0,0},Table[Binomial[n,2]BellB[n-2],{n,2,30}]] (* _Harvey P. Dale_, May 06 2014 *)
%o (Python)
%o from itertools import count, accumulate, islice
%o def A105479_gen(): # generator of terms
%o yield from (0,0,1)
%o blist, b, c = (1,), 1, 1
%o for n in count(2):
%o c += n
%o blist = list(accumulate(blist, initial=(b:=blist[-1])))
%o yield b*c
%o A105479_list = list(islice(A105479_gen(),20)) # _Chai Wah Wu_, Jun 22 2022
%Y Cf. A105480, A105489, A105484, A124498.
%Y Column k=2 of A193297.
%K easy,nonn
%O 0,4
%A _Augustine O. Munagi_, Apr 10 2005
%E Edited by _N. J. A. Sloane_, Jan 01 2007
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