A076822 Number of partitions of the n-th triangular number involving only the numbers 1..n and with exactly n terms.

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

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

a(n) is the number of fair dice with n sides and expected value (n+1)/2 with distinct composition of numbers between 1 and n. - Felix Huber, Aug 02 2024

Links

Max Alekseyev and Alois P. Heinz, Table of n, a(n) for n = 0..240 (terms n=1..100 from Max Alekseyev)

L. Takács, Some asymptotic formulas for lattice paths, J. Statist. Plann. Inference, 14 (1986), 123-142.

Formula

a(n) = A067059(n,n+1); also a(n) = T[n*(n-1)/2, n-1, n] with T[ ] defined as in A047993. - Martin Fuller, Jun 27 2006

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

Cf. A067059, A047993, A039744.

Cf. A002838. [From R. J. Mathar, Sep 20 2008]

Cf. A188181 (columns 1, 2).

Sequence in context: A218781 A363682 A002838 * A143657 A014326 A148284

Adjacent sequences: A076819 A076820 A076821 * A076823 A076824 A076825

Keyword

nonn

Author

Jon Perry, Nov 19 2002

Extensions

Edited and extended to 12 terms by Robert G. Wilson v, Nov 23 2002

Further terms from Max Alekseyev, May 24 2007

a(0)=1 prepended by Alois P. Heinz, May 28 2016

Status

approved