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Number of partitions of triangular numbers n*(n+1)/2 into (n-2) distinct parts for n>=3.
4

%I #18 Aug 01 2020 11:38:57

%S 1,4,12,27,57,110,201,352,598,984,1586,2503,3882,5928,8932,13287,

%T 19551,28472,41078,58754,83372,117417,164230,228212,315190,432817,

%U 591130,803192,1086035,1461680,1958596,2613417,3473190,4598073,6064920,7971480

%N Number of partitions of triangular numbers n*(n+1)/2 into (n-2) distinct parts for n>=3.

%C In triangle A104382, equals the second diagonal down from the main diagonal.

%C Also equals a diagonal with slope -3 in the Partition Numbers triangle A008284, found at n = 3+3k, or T(3+3k,k) for k >=1. - _Richard R. Forberg_, Dec 02 2014

%H Vaclav Kotesovec, <a href="/A104384/b104384.txt">Table of n, a(n) for n = 3..132</a>

%F From _Álvar Ibeas_, Jul 23 2020: (Start)

%F Writing p(m) for A000041(m),

%F a(n) = p(2n-1) - A000070(n) + 1 and

%F a(n+1) - a(n) = p(2*n+1) - p(2*n-1) - p(n+1) = A027336(2*n+1) - p(n+1).

%F (End)

%o (PARI) {a(n)=if(n<3,0,polcoeff(polcoeff( prod(i=1,n*(n+1)/2,1+y*x^i,1+x*O(x^(n*(n+1)/2))),n*(n+1)/2,x),n-2,y))}

%Y Cf. A000009, A104382, A008284.

%K nonn

%O 3,2

%A _Paul D. Hanna_, Mar 04 2005