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%I #12 Jul 10 2020 22:07:40
%S 1,5,30,175,1050,6552,42630,289410,2049300,15120105,116090975,
%T 926248050,7668746540,65793760060,584151925320,5360347320420,
%U 50776288702215,495946245776940,4989391837053085,51648932225779735,549620905409062872
%N Number of partitions of {1...n} containing 4 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.
%D A. O. Munagi, Set Partitions with Successions and Separations, Int. J. Math and Math. Sc. 2005, no. 3 (2005), 451-463.
%H A. O. Munagi, <a href="http://www.emis.de/journals/HOA/IJMMS/2005/3451.pdf">Set Partitions with Successions and Separations</a>, IJMMS 2005:3 (2005), 451-463.
%F a(n) = binomial(n-1, 4)*Bell(n-5), the case r = 4 in the general case of r pairs: c(n, r) = binomial(n-1, r)*B(n-r-1).
%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>=4, a(n+1)=(-1)^(n-4)*coeff(charpoly(A,x),x^4). [_Milan Janjic_, Jul 08 2010]
%F E.g.f.: (1/4!) * Integral (x^4 * exp(exp(x) - 1)) dx. - _Ilya Gutkovskiy_, Jul 10 2020
%e a(6) = 5 because the partitions of {1,2,3,4,5,6} with 4 pairs of consecutive integers are 12345/6,1234/56,123/456,12/3456,1/23456.
%p seq(binomial(n-1,4)*combinat[bell](n-5),n=5..25);
%Y Cf. A105480, A105482, A105486, A105491, A105494.
%K easy,nonn
%O 5,2
%A _Augustine O. Munagi_, Apr 10 2005
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