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%I #11 May 27 2026 00:27:37
%S 0,1,1,4,29,336,5687,132293,4047932,157599498,7607014464,445789277010,
%T 31176975417126,2564948607907686,245220679951969722,
%U 26959156361305448280,3377230301505301128888,478235430296927841409872,76010484446415665803480824,13474613251341094194492053904
%N The total number of at-most-6-furcating labeled histories that can be constructed for n labeled leaves.
%C a(n) is the number of "merger histories" of n elements, where at most 6 elements can merge at the same time (see A256006).
%H Emily H. Dickey and Noah A. Rosenberg, <a href="https://doi.org/10.1016/j.dam.2026.04.032">Labeled histories and maximally probable labeled topologies with multifurcation</a>, Discr. Appl. Math. 391 (2026), 192-203. See Table 1.
%F a(n) = (n*(n-1)/2)*a(n-1) + (n*(n-1)*(n-2)/6)*a(n-2) + (n*(n-1)*(n-2)*(n-3)/24)*a(n-3) + (n*(n-1)*(n-2)*(n-3)*(n-4)/120)*a(n-4) + (n*(n-1)*(n-2)*(n-3)*(n-4)*(n-5)/720)*a(n-5) for n>=6; base cases follow A256006(n-1) for 1<=n<=5.
%t RecurrenceTable[{A[n] == Binomial[n,2] A[n-1] + Binomial[n,3] A[n-2] + Binomial[n,4] A[n-3] + Binomial[n,5] A[n-4] + Binomial[n,6] A[n-5], A[0]==0, A[1]==1, A[2]==1, A[3]==4, A[4]==29}, A, {n, 20}]
%Y Cf. A006472, A256006, A358072, A396065, A393749.
%K nonn
%O 0,4
%A _Noah A Rosenberg_, May 21 2026