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A158497 Triangle T(n,k) formed by the coordination sequences and the number of leaves for trees. 1

%I

%S 1,1,1,1,2,2,1,3,6,12,1,4,12,36,108,1,5,20,80,320,1280,1,6,30,150,750,

%T 3750,18750,1,7,42,252,1512,9072,54432,326592,1,8,56,392,2744,19208,

%U 134456,941192,6588344

%N Triangle T(n,k) formed by the coordination sequences and the number of leaves for trees.

%C Consider the k-fold Cartesian products CP(n,k) of the vector A(n)=[1,2,3,...,n].

%C An element of CP(n,k) is a n-tuple T_t of the form T_t=[i_1,i_2,i_3,...,i_k] with t=1,..,n^k.

%C We count members T of CP(n,k) which satisfy some condition delta(T_t), so delta(.) is an indicator function which attains values of 1 or 0 depending on whether T_t is to be counted or not; the summation sum_{CP(n,k)} delta(T_t) over all elements T_t of CP produces the count.

%C For the triangle here we have delta(T_t) = 0 if for any two i_j, i_(j+1) in T_t one has i_j = i_(j+1): T(n,k) = Sum_{CP(n,k)} delta(T_t) = Sum_{CP(n,k)} delta(i_j = i_(j+1)).

%C The test on i_j > i_(j+1) generates A158498. One gets the Pascal triangle A007318 if the indicator function tests whether for any two i_j, i_(j+1) in T_t one has i_j >= i_(j+1).

%C Use of other indicator functions can also calculate the Bell numbers A000110, A000045 or A000108.

%F Columns: T(n,2)=A002378(n-1). T(n,3)=A011379(n-1). T(n,5)=A101362(n-1).

%F Rows: T(2,k) = A040000(k). T(3,k) = A003945(k), T(4,k)=A003946(k), T(5,k)=A003947(k), T(6,k)=A003948(k).

%F T(n,k) = (n-1)^(k-1)+(n-1)^k = n*A079901(n-1,k-1), k>0.

%F sum_{k=0..n} T(n,k) = (n*(n-1)^n-2)/(n-2), n>2.

%e The triangle begins

%e 1

%e 1 1

%e 1 2 2

%e 1 3 6 12

%e 1 4 12 36 108

%e 1 5 20 80 320 1280

%e 1 6 30 150 750 3750 18750

%e 1 7 42 252 1512 9072 54432 326592

%e 1 8 56 392 2744 19208 134456 941192 6588344

%e T(3,3)=12 counts the triples (1,2,1), (1,2,3), (1,3,1), (1,3,2), (2,1,2), (2,1,3), (2,3,1), (2,3,2), (3,1,2), (3,1,3), (3,2,1), (3,2,3) out of a total of 3^3=27 triples in the CP(3,3).

%Y Cf. A007318, A158498.

%K nonn,tabl

%O 0,5

%A _Thomas Wieder_, Mar 20 2009

%E Edited by _R. J. Mathar_, Mar 31 2009

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Last modified September 20 20:02 EDT 2020. Contains 337265 sequences. (Running on oeis4.)