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A214576
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Triangle read by rows: T(n,k) is the number of partitions of n in which each part is divisible by the next and have last part equal to k (1<=k<=n).
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2
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1, 1, 1, 2, 0, 1, 3, 1, 0, 1, 5, 0, 0, 0, 1, 6, 2, 1, 0, 0, 1, 10, 0, 0, 0, 0, 0, 1, 11, 3, 0, 1, 0, 0, 0, 1, 16, 0, 2, 0, 0, 0, 0, 0, 1, 19, 5, 0, 0, 1, 0, 0, 0, 0, 1, 26, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 27, 6, 3, 2, 0, 1, 0, 0, 0, 0, 0, 1, 40, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1
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OFFSET
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1,4
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COMMENTS
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T(n,k) is also the number of generalized Bethe trees with n edges and root degree k.
A generalized Bethe tree is a rooted tree in which vertices at the same level have the same degree; they are called uniform trees in the Goldberg and Livshits reference.
There is a simple bijection between generalized Bethe trees with n edges and partitions of n in which each part is divisible by the next (the parts are given by the number of edges at the successive levels). We have the correspondences: root degree --- last part; number of leaves --- first part; height --- number of parts.
Sum of entries in row n is A003238(n+1).
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LINKS
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M. K. Goldberg and E. M. Livshits, On minimal universal trees, Mathematical Notes of the Acad. of Sciences of the USSR, 4, 1968, 713-717 (translation from the Russian Mat. Zametki 4 1968 371-379).
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FORMULA
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T(n,k)=a(n/k -1) if k|n and = 0 otherwise; here a(n) is defined by a(0)=1, a(n) = sum_{j|n}a(j-1). We have a(n) = A003238(n+1) = number of partitions of n in which each part is divisible by the next one.
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EXAMPLE
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T(9,3)=2 because we have (6,3) and (3,3,3).
Triangle starts:
1;
1, 1;
2, 0, 1;
3, 1, 0, 1;
5, 0, 0, 0, 1;
6, 2, 1, 0, 0, 1;
10, 0, 0, 0, 0, 0, 1;
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MAPLE
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with(numtheory): a := proc (n) if n = 0 then 1 else add(a(divisors(n)[j]-1), j = 1 .. tau(n)) end if end proc: T := proc (n, k) if type(n/k, integer) = true then a(n/k-1) else 0 end if end proc: for n to 18 do seq(T(n, k), k = 1 .. n) end do; # yields sequence in triangular form
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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