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

Author

Jon Perry, Nov 19 2002

Keywords

• nonn

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

Examples

			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.
		

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.

Crossrefs

Cf. A067059, A047993, A039744.

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

Cf. A188181 (columns 1, 2).

Programs

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+', "
")';
    eval(str);
    ccc[cccc++ ]=cc;
    document.print('****
');
}
document.write(ccc);

Mathematica

f[n_] := Block[{p = IntegerPartitions[n(n + 1)/2, n]}, Length[ Select[p, Length[ # ] == n &]]]; Table[ f[n], {n, 1, 13}]

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

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