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A158497
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Triangle T(n,k) formed by the coordination sequences and the number of leaves for trees.
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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, 3750, 18750, 1, 7, 42, 252, 1512, 9072, 54432, 326592, 1, 8, 56, 392, 2744, 19208, 134456, 941192, 6588344
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 0,5
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COMMENTS
| Consider the k-fold Cartesian products CP(n,k) of the vector A(n)=[1,2,3,...,n].
An element of CP(n,k) is a n-tupel T_t of the form T_t=[i_1,i_2,i_3,...,i_k] with t=1,..,n^k.
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.
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)).
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).
Use of other indicator functions can also calculate the Bell numbers A000110, A000045 or A000108.
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FORMULA
| Columns: T(n,2)=A002378(n-1). T(n,3)=A011379(n-1). T(n,5)=A101362(n-1).
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).
T(n,k) = (n-1)^(k-1)+(n-1)^k = n*A079901(n-1,k-1), k>0.
sum_{k=0..n} T(n,k) = (n*(n-1)^n-2)/(n-2), n>2.
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EXAMPLE
| The triangle begins
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 3750 18750
1 7 42 252 1512 9072 54432 326592
1 8 56 392 2744 19208 134456 941192 6588344
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).
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CROSSREFS
| A007318, A158498.
Sequence in context: A008279 A056043 A187005 * A110564 A007441 A111933
Adjacent sequences: A158494 A158495 A158496 * A158498 A158499 A158500
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KEYWORD
| nonn,tabl
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AUTHOR
| Thomas Wieder (thomas.wieder(AT)t-online.de), Mar 20 2009
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EXTENSIONS
| Edited by R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Mar 31 2009
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