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A007713
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Number of 4-level rooted trees with n leaves.
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17
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1, 1, 4, 10, 30, 75, 206, 518, 1344, 3357, 8429, 20759, 51044, 123973, 299848, 719197, 1716563, 4070800, 9607797, 22555988, 52718749, 122655485, 284207304, 655894527, 1508046031, 3454808143, 7887768997, 17949709753, 40719611684, 92096461012, 207697731344
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OFFSET
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0,3
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LINKS
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B. A. Huberman and T. Hogg, Complexity and adaptation, Evolution, games and learning (Los Alamos, N.M., 1985). Phys. D 22 (1986), no. 1-3, 376-384.
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FORMULA
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Euler transform applied thrice to all-1's sequence.
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EXAMPLE
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Also the number of multiset partitions of multiset partitions of integer partitions of n. For example, the a(1) = 1 through a(4) = 30 multiset partitions are:
((1)) ((2)) ((3)) ((4))
((11)) ((12)) ((13))
((1)(1)) ((111)) ((22))
((1))((1)) ((1)(2)) ((112))
((1)(11)) ((1111))
((1))((2)) ((1)(3))
((1))((11)) ((2)(2))
((1)(1)(1)) ((1)(12))
((1))((1)(1)) ((2)(11))
((1))((1))((1)) ((1)(111))
((11)(11))
((1))((3))
((2))((2))
((1))((12))
((1)(1)(2))
((2))((11))
((1))((111))
((1)(1)(11))
((11))((11))
((1))((1)(2))
((2))((1)(1))
((1))((1)(11))
((1)(1)(1)(1))
((11))((1)(1))
((1))((1))((2))
((1))((1))((11))
((1))((1)(1)(1))
((1)(1))((1)(1))
((1))((1))((1)(1))
((1))((1))((1))((1))
(End)
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MAPLE
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with(numtheory): etr:= proc(p) local b; b:=proc(n) option remember; local d, j; if n=0 then 1 else add(add(d*p(d), d=divisors(j)) *b(n-j), j=1..n)/n fi end end: b0:= etr(1): b1:= etr(b0): a:= etr(b1): seq(a(n), n=0..30); # Alois P. Heinz, Sep 08 2008
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MATHEMATICA
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i[ n_, m_ ] := 1 /; m==1 || n==0; i[ n_, m_ ] := (i[ n, m ]=1/n Sum[ i[ k, m ] Plus @@ ((# i[ #, m-1 ])& /@ Divisors[ n-k ]), {k, 0, n-1} ]) /; n>0 && m>1
etr[p_] := Module[{b}, b[n_] := b[n] = If[n == 0, 1, Sum[Sum[d*p[d], {d, Divisors[ j]}]*b[n-j], {j, 1, n}]/n]; b]; b0 = etr[Function[1]]; b1 = etr[b0]; a = etr[b1]; Table[a[n], {n, 1, 30}] (* Jean-François Alcover, Mar 05 2015, after Alois P. Heinz *)
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CROSSREFS
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Cf. A001970, A047968, A050342, A089259, A141268, A258466, A261049, A319066, A320328, A320330, A320331.
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KEYWORD
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easy,nonn
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AUTHOR
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STATUS
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approved
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