|
|
A076822
|
|
Number of partitions of the n-th triangular number involving only the numbers 1..n and with exactly n terms.
|
|
6
|
|
|
1, 1, 1, 2, 5, 12, 32, 94, 289, 910, 2934, 9686, 32540, 110780, 381676, 1328980, 4669367, 16535154, 58965214, 211591218, 763535450, 2769176514, 10089240974, 36912710568, 135565151486, 499619269774, 1847267563742, 6850369296298
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
0,4
|
|
COMMENTS
|
Asymptotic to (sqrt(3)/(2*Pi))*(4^n/n^2). It is the number of lattice paths from (0,0) to (n,n-1) with steps only to the right or upward and having area n(n-1)/2 between the path and the x-axis. In the reference by Takács use formula (77) with a=n, b=n(n-1)/2 and then Stirling's formula. - Kent E. Morrison, May 28 2016
|
|
LINKS
|
|
|
FORMULA
|
|
|
EXAMPLE
|
a(4)=5 as T(4)=10= 1+1+4+4 =1+2+3+4 = 1+3+3+3 = 2+2+2+4 = 2+2+3+3.
|
|
MATHEMATICA
|
f[n_] := Block[{p = IntegerPartitions[n(n + 1)/2, n]}, Length[ Select[p, Length[ # ] == n &]]]; Table[ f[n], {n, 1, 13}]
|
|
PROG
|
(JavaScript)
ccc=new Array(); cccc=0;
for (n=1; n<11; n++)
{
str='cc=0; for (i1=1; i1<'+(n+1)+'; i1++)';
str2='i1';
str3='i1';
tn=1;
for (i=2; i<=n; i++)
{
str+='for (i'+i+'=i'+(i-1)+'; i'+i+'<'+(n+1)+'; i'+i+'++)';
str2+='+i'+i;
str3+=', ", ", i'+i;
tn+=i;
}
str+='if ('+str2+'=='+tn+') document.print(++cc, ":", '+str3+', "<br>")';
eval(str);
ccc[cccc++ ]=cc;
document.print('****<br>');
}
document.write(ccc);
|
|
CROSSREFS
|
|
|
KEYWORD
|
nonn
|
|
AUTHOR
|
|
|
EXTENSIONS
|
|
|
STATUS
|
approved
|
|
|
|