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 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 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+', "
")';     eval(str);     ccc[cccc++ ]=cc;     document.print('****
'); } 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: A148283 A218781 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

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Last modified October 18 20:42 EDT 2019. Contains 328197 sequences. (Running on oeis4.)