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A214576 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).

%I A214576 #21 Jan 30 2022 11:41:23
%S A214576 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,
%T A214576 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,
%U A214576 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
%N A214576 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).
%C A214576 T(n,k) is also the number of generalized Bethe trees with n edges and root degree k.
%C A214576 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.
%C A214576 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.
%C A214576 Sum of entries in row n is A003238(n+1).
%H A214576 Seiichi Manyama, <a href="/A214576/b214576.txt">Rows n = 1..140, flattened</a>
%H A214576 M. K. Goldberg and E. M. Livshits, <a href="http://mi.mathnet.ru/eng/mz9458">On minimal universal trees</a>, Mathematical Notes of the Acad. of Sciences of the USSR, 4, 1968, 713-717 (translation from the Russian Mat. Zametki 4 1968 371-379).
%H A214576 O. Rojo, <a href="http://dx.doi.org/10.1016/j.laa.2008.01.026">Spectra of weighted generalized Bethe trees joined at the root</a>, Linear Algebra and its Appl., 428, 2008, 2961-2979.
%F A214576 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.
%e A214576 T(9,3)=2 because we have (6,3) and (3,3,3).
%e A214576 Triangle starts:
%e A214576    1;
%e A214576    1, 1;
%e A214576    2, 0, 1;
%e A214576    3, 1, 0, 1;
%e A214576    5, 0, 0, 0, 1;
%e A214576    6, 2, 1, 0, 0, 1;
%e A214576   10, 0, 0, 0, 0, 0, 1;
%p A214576 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
%Y A214576 Cf. A003238, A122934, A214575.
%K A214576 nonn,tabl
%O A214576 1,4
%A A214576 _Emeric Deutsch_, Aug 18 2012