cp's OEIS Frontend

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

%I A076822 #54 Aug 17 2024 23:26:31
%S A076822 1,1,1,2,5,12,32,94,289,910,2934,9686,32540,110780,381676,1328980,
%T A076822 4669367,16535154,58965214,211591218,763535450,2769176514,10089240974,
%U A076822 36912710568,135565151486,499619269774,1847267563742,6850369296298
%N A076822 Number of partitions of the n-th triangular number involving only the numbers 1..n and with exactly n terms.
%C A076822 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
%C A076822 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
%H A076822 Max Alekseyev and Alois P. Heinz, <a href="/A076822/b076822.txt">Table of n, a(n) for n = 0..240</a> (terms n=1..100 from Max Alekseyev)
%H A076822 L. Takács, <a href="https://dx.doi.org/10.1016/0378-3758(86)90016-9">Some asymptotic formulas for lattice paths</a>, J. Statist. Plann. Inference, 14 (1986), 123-142.
%F A076822 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
%e A076822 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.
%t A076822 f[n_] := Block[{p = IntegerPartitions[n(n + 1)/2, n]}, Length[ Select[p, Length[ # ] == n &]]]; Table[ f[n], {n, 1, 13}]
%o A076822 (JavaScript)
%o A076822 ccc=new Array(); cccc=0;
%o A076822 for (n=1; n<11; n++)
%o A076822 {
%o A076822     str='cc=0; for (i1=1; i1<'+(n+1)+'; i1++)';
%o A076822     str2='i1';
%o A076822     str3='i1';
%o A076822     tn=1;
%o A076822     for (i=2; i<=n; i++)
%o A076822     {
%o A076822         str+='for (i'+i+'=i'+(i-1)+'; i'+i+'<'+(n+1)+'; i'+i+'++)';
%o A076822         str2+='+i'+i;
%o A076822         str3+=', ", ", i'+i;
%o A076822         tn+=i;
%o A076822     }
%o A076822     str+='if ('+str2+'=='+tn+') document.print(++cc, ":", '+str3+', "<br>")';
%o A076822     eval(str);
%o A076822     ccc[cccc++ ]=cc;
%o A076822     document.print('****<br>');
%o A076822 }
%o A076822 document.write(ccc);
%Y A076822 Cf. A067059, A047993, A039744.
%Y A076822 Cf. A002838. [From _R. J. Mathar_, Sep 20 2008]
%Y A076822 Cf. A188181 (columns 1, 2).
%K A076822 nonn
%O A076822 0,4
%A A076822 _Jon Perry_, Nov 19 2002
%E A076822 Edited and extended to 12 terms by _Robert G. Wilson v_, Nov 23 2002
%E A076822 Further terms from _Max Alekseyev_, May 24 2007
%E A076822 a(0)=1 prepended by _Alois P. Heinz_, May 28 2016