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%I #12 Jul 26 2017 03:15:07
%S 2,20,150,1040,7105,49112,347760,2537640,19135875,149285400,
%T 1205088742,10062575068,86859191510,774456785200,7126496659960,
%U 67617733760064,660932425168071,6649326113764980,68793130453044760,731299516881396540
%N Number of partitions of {1...n} containing 3 detached pairs of consecutive integers, i.e., partitions in which only 1- or 2-strings of consecutive integers can appear in a block and there are exactly three 2-strings.
%C Number of partitions enumerated by A105480 in which the maximal length of consecutive integers in a block is 2.
%C With offset 3t, number of partitions of {1...N} containing 3 detached strings of t consecutive integers, where N = n + 3j, t = 2 + j, j = 0, 1, 2, ..., i.e., partitions of {1,..,N} in which only v-strings of consecutive integers can appear in a block, where v=1 or v=t and there are exactly three t-strings.
%H A. 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-3, 3)*Bell(n-4), which is the case r=3 in the general case of r pairs, d(n,r) = binomial(n-r, r)*Bell(n-r-1), which is the case t=2 of the general formula d(n,r,t) = binomial(n-r*(t-1), r)*Bell(n-r*(t-1)-1).
%e a(6) = 2 because the partitions of {1,2,3,4,5,6} with 3 detached pairs of consecutive integers are 12/34/56, 1256/34.
%p seq(binomial(n-3,3)*combinat[bell](n-4),n=6..25);
%p a:=n->sum(numbcomb (n,2)*bell(n)/3, j=0..n): seq(a(n), n=2..21); # _Zerinvary Lajos_, Apr 25 2007
%Y Cf. A105480, A105485, A105488, A105490.
%K easy,nonn
%O 6,1
%A _Augustine O. Munagi_, Apr 10 2005
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