back to the attic
c(n,k) Triangles
An original result
References taken from the Encyclopedia of Integer Sequences
: A008280
- A008281
- A008282
Around
The above formula is the analytical expression of the *c(n,k)*
definition.
(in this case, we have *k=n*)
But in *1966*, Entringer was the first to make the following
table which records the spread of the up-down permutation following the
value of their first terms.
To make it, make the integer *c(n,k) *triangle, *(0kn)* in which each integer *c(n,k)* (
not to be confuse with combination (n
choose k) ) is the sum of the
last *k* terms from the *n-1*th line :
*c(n,k)* |
*k=0* |
*1* |
*2* |
*3* |
*4* |
*5* |
*6* |
*7* |
*8* |
*n=0* |
*1* |
* * |
* * |
* * |
* * |
* * |
* * |
* * |
* * |
*1* |
*0* |
*1* |
* * |
* * |
* * |
* * |
* * |
* * |
* * |
*2* |
*0* |
*1* |
*1* |
* * |
* * |
* * |
* * |
* * |
* * |
*3* |
*0* |
*1* |
*2* |
*2* |
* * |
* * |
* * |
* * |
* * |
*4* |
*0* |
*2* |
*4* |
*5* |
*5* |
* * |
* * |
* * |
* * |
*5* |
*0* |
*5* |
*10* |
*14* |
*16* |
*16* |
* * |
* * |
* * |
*6* |
*0* |
*16* |
*32* |
*46* |
*56* |
*61* |
*61* |
* * |
* * |
*7* |
*0* |
*61* |
*122* |
*178* |
*224* |
*256* |
*272* |
*272* |
* * |
*8* |
*0* |
*272* |
*544* |
*800* |
*1024* |
*1202* |
*1324* |
*1385* |
*1385* |
The numbers *c*_{n} seemed to have been known since Euler and comply to the definition at the top of
the page.
In *1879*, Désiré André has found the
property to be also the number of up-down permutations of the interger *1*
to *n* (this means that the *n-1* successive difference
between two consecutif terms of the permutation alternate between
positive and negative.).
For exemple, *c*_{4}=5 and we do have 5 up-down
permutation of *{1,2,3,4}* :
*1 3 2 4**1 4 2 3**2 3 1 4**2 4 1 3**3 4 1 2*
For *1 3 2 4*, we do have *3-1=2*, *2-3=-1*, *4-2=2*
Notice that *c*_{n}=c(n,n). we then get
Method of prove
In fact, we can show that a power serie with general
term gives */2* for convergence, (which
seems logival if we consider the error bound in the Taylor expansion of
the funtion f defined above)
Hence, the ration of the *z*^{n} coefficients must tend
towards */2* according to Alembert's result.
Trials
The fraction give a near value of Pi by default if n is even
and by excess otherwise :
*n=0* |
*1* |
*2* |
*3* |
*4* |
*5* |
*6* |
*7* |
*8* |
*9* |
*10* |
*2* |
*4* |
*3* |
*3,2* |
*3,125* |
*3,147* |
*3,1397* |
*3,1422* |
*3,14138* |
*3,1416* |
*3,141569* |
It seems to me that it converges logarithmiticaly, but I don't have the
proof...
retour au grenier back to home page |