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A066411
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Form a triangle with the numbers [0..n] on the base, where each number is the sum of the two below; a(n) is the number of different possible values for the apex.
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7
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1, 1, 3, 5, 23, 61, 143, 215, 995, 2481, 5785, 12907, 29279, 64963, 144289, 158049, 683311, 1471123, 3166531, 6759177, 14404547, 30548713
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OFFSET
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0,3
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COMMENTS
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a(n) is the number of different possible sums of c_k * (n choose k) where the c_k are a permutation of 0 through n. - Joshua Zucker, May 08 2006
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LINKS
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Table of n, a(n) for n=0..21.
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EXAMPLE
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For n = 2 we have three triangles:
..4.......5.......3
.1,3.....2,3.....2,1
0,1,2...0,2,1...2,0,1
with three different values for the apex, so a(2) = 3.
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MATHEMATICA
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g[s_List] := Plus @@@ Partition[s, 2, 1]; f[n_] := Block[{k = 1, lmt = 1 + (n + 1)!, lst = {}, p = Permutations[Range[0, n]]}, While[k < lmt, AppendTo[ lst, Nest[g, p[[k]], n][[1]]]; k++]; lst]; Table[ Length@ Union@ f@ n, {n, 0, 10}] (* Robert G. Wilson v, Jan 24 2012 *)
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PROG
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(MATLAB)
for n=0:9
size(unique(perms(0:n)*diag(fliplr(pascal(n+1)))), 1)
end % Nathaniel Johnston, Apr 20 2011
(C++)
#include <iostream>
#include <vector>
#include <set>
#include <algorithm>
using namespace std;
inline long long pascApx(const vector<int> & s)
{
const int n = s.size() ;
vector<long long> scp(n) ;
for(int i=0; i<n; i++)
scp[i] = s[i] ;
for(int i=1; i<n; i++)
for(int acc=0 ; acc < n-i ; acc++)
scp[acc] += scp[acc+1] ;
return scp[0] ;
}
int main(int argc, char *argv[])
{
for(int n=1 ; ; n++)
{
vector<int> s;
for(int i=0; i<n; i++)
s.push_back(i) ;
set<long long> apx;
do
{
apx.insert( pascApx(s)) ;
} while( next_permutation(s.begin(), s.end()) ) ;
cout << n << " " << apx.size() << endl ;
}
return 0 ;
} /* R. J. Mathar, Jan 24 2012 */
(PARI) A066411(n)={my(u=0, o=A189391(n), v, b=vector(n++, i, binomial(n-1, i-1))~); sum(k=1, n!\2, !bittest(u, numtoperm(n, k)*b-o) & u+=1<<(numtoperm(n, k)*b-o))} \\ M. F. Hasler, Jan 24 2012
(Haskell)
import Data.List (permutations, nub)
a066411 0 = 1
a066411 n = length $ nub $ map
apex [perm | perm <- permutations [0..n], head perm < last perm] where
apex = head . until ((== 1) . length)
(\xs -> (zipWith (+) xs $ tail xs))
-- Reinhard Zumkeller, Jan 24 2012
(Python)
from sympy import binomial
def partitionpairs(xlist): # generator of all partitions into pairs and at most 1 singleton, returning the sums of the pairs
if len(xlist) <= 2:
yield [sum(xlist)]
else:
m = len(xlist)
for i in range(m-1):
for j in range(i+1, m):
rem = xlist[:i]+xlist[i+1:j]+xlist[j+1:]
y = [xlist[i]+xlist[j]]
for d in partitionpairs(rem):
yield y+d
def A066411(n):
b = [binomial(n, k) for k in range(n//2+1)]
return len(set((sum(d[i]*b[i] for i in range(n//2+1)) for d in partitionpairs(list(range(n+1)))))) # Chai Wah Wu, Oct 19 2021
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CROSSREFS
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Cf. A062684, A062896, A099325, A189162, A189390 (maximum apex value), A189391 (minimum apex value).
Sequence in context: A290384 A023247 A027753 * A153410 A230080 A155778
Adjacent sequences: A066408 A066409 A066410 * A066412 A066413 A066414
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KEYWORD
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nonn,nice,more
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AUTHOR
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Naohiro Nomoto, Dec 25 2001
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EXTENSIONS
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More terms from John W. Layman, Jan 07 2003
a(10) from Nathaniel Johnston, Apr 20 2011
a(11) from Alois P. Heinz, Apr 21 2011
a(12) and a(13) from Joerg Arndt, Apr 21 2011
a(14)-a(15) from Alois P. Heinz, Apr 27 2011
a(0)-a(15) verified by R. H. Hardin Jan 27 2012
a(16) from Alois P. Heinz, Jan 28 2012
a(17)-a(21) from Graeme McRae, Jan 28, Feb 01 2012
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STATUS
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approved
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