A002060 Number of partitions of an n-gon into (n-5) parts.

4, 60, 550, 4004, 25480, 148512, 813960, 4263600, 21573816, 106234700, 511801290, 2421810300, 11289642000, 51967090560, 236635858800, 1067518772640, 4776759725400, 21221827263000, 93687293423724, 411270420524040, 1796296260955504, 7809983743284800, 33816739954270000

Offset: 7

Author

N. J. A. Sloane

Keywords

• nonn

Comments

a(n) = V(r=n,k=n-5), 4th subdiagonal of the triangle of V on page 240.

It appears that V(r=15,k=10) in the Cayley table is an error, so the sequence was intended to be 4, 60, 550, 4004, 25480, 148512, 813960, 4263600, 21573816, 106234700, 511801290, 2421810300, 11289642000, 51967090560, 236635858800... - R. J. Mathar, Nov 26 2011

References

• N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• A. Cayley, On the partitions of a polygon, Proc. London Math. Soc., 22 (1891), 237-262 = Collected Mathematical Papers. Vols. 1-13, Cambridge Univ. Press, London, 1889-1897, Vol. 13, pp. 93ff.

Crossrefs

Cf. A002058, A002059.

Programs

Maple

V := proc(r,k)
        local a ,t;
        a := k-1;
        for t from k-2 to 1 by -1 do
                a := a*(r+t)/(t+2) ;
        end do:
        for t from 3 to k+1 do
                a := a*(r-t)/(k-t+2) ;
        end do:
        a ;
end proc:
A002060 := proc(n)
        V(n,n-5) ;
end proc:
seq(A002060(n),n=7..25) ; # R. J. Mathar, Nov 26 2011

Mathematica

V[r_, k_] := Module[{a, t}, a = k - 1;
   For[t = k - 2, t >= 1, t--, a = a*(r + t)/(t + 2)];
   For[t = 3, t <= k + 1, t++, a = a*(r - t)/(k - t + 2)];
   a];
A002060[n_] := V[n, n - 5];
Table[A002060[n], {n, 7, 29}] (* Jean-François Alcover, Mar 10 2023, after R. J. Mathar *)

Extensions

More terms from Hugo Pfoertner, Dec 26 2021