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

%I A002060 M3691 N1509 #28 Mar 10 2023 03:18:38
%S A002060 4,60,550,4004,25480,148512,813960,4263600,21573816,106234700,
%T A002060 511801290,2421810300,11289642000,51967090560,236635858800,
%U A002060 1067518772640,4776759725400,21221827263000,93687293423724,411270420524040,1796296260955504,7809983743284800,33816739954270000
%N A002060 Number of partitions of an n-gon into (n-5) parts.
%C A002060 a(n) = V(r=n,k=n-5), 4th subdiagonal of the triangle of V on page 240.
%C A002060 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
%D A002060 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D A002060 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A002060 A. Cayley, <a href="http://dx.doi.org/10.1112/plms/s1-22.1.237">On the partitions of a polygon</a>, Proc. London Math. Soc., 22 (1891), 237-262 = Collected Mathematical Papers. Vols. 1-13, Cambridge Univ. Press, London, 1889-1897, Vol. 13, pp. 93ff.
%p A002060 V := proc(r,k)
%p A002060         local a ,t;
%p A002060         a := k-1;
%p A002060         for t from k-2 to 1 by -1 do
%p A002060                 a := a*(r+t)/(t+2) ;
%p A002060         end do:
%p A002060         for t from 3 to k+1 do
%p A002060                 a := a*(r-t)/(k-t+2) ;
%p A002060         end do:
%p A002060         a ;
%p A002060 end proc:
%p A002060 A002060 := proc(n)
%p A002060         V(n,n-5) ;
%p A002060 end proc:
%p A002060 seq(A002060(n),n=7..25) ; # _R. J. Mathar_, Nov 26 2011
%t A002060 V[r_, k_] := Module[{a, t}, a = k - 1;
%t A002060    For[t = k - 2, t >= 1, t--, a = a*(r + t)/(t + 2)];
%t A002060    For[t = 3, t <= k + 1, t++, a = a*(r - t)/(k - t + 2)];
%t A002060    a];
%t A002060 A002060[n_] := V[n, n - 5];
%t A002060 Table[A002060[n], {n, 7, 29}] (* _Jean-François Alcover_, Mar 10 2023, after _R. J. Mathar_ *)
%Y A002060 Cf. A002058, A002059.
%K A002060 nonn
%O A002060 7,1
%A A002060 _N. J. A. Sloane_
%E A002060 More terms from _Hugo Pfoertner_, Dec 26 2021