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%I #25 Mar 16 2024 13:20:45
%S 0,0,0,0,1,1,2,3,6,5,10,12,16,21,28,30,45,53,63,81,99,114,144,173,204,
%T 245,293,340,410,482,554,662,774,890,1044,1207,1393,1619,1864,2134,
%U 2464,2828,3220,3701,4223,4789,5474,6223,7050,8004,9058,10230,11579
%N Number of partitions p of n such that (number of distinct parts of p) = max(p) - min(p).
%H John Tyler Rascoe, <a href="/A239956/b239956.txt">Table of n, a(n) for n = 0..100</a>
%F A239954(n) + a(n) + A034296(n) = A000041(n) for n >= 1.
%F G.f.: Sum_{m>0} A(x,m), where A(x,m) = Sum_{i>m+1} x^(i+m)/((1-x^i)*(1-x^m)) * Sum_{j=m+1..i-1} ( (1-x^j)/(x^j) * Product_{k=m+1..i-1} (x^k/(1-x^k)) ) is the g.f. for partitions of this kind with min(p) = m. - _John Tyler Rascoe_, Mar 16 2024
%e a(8) counts these 6 partitions: 53, 431, 422, 4211, 3311, 311111.
%t z = 60; d[p_] := d[p] = Length[DeleteDuplicates[p]]; f[p_] := f[p] = Max[p] - Min[p]; g[n_] := g[n] = IntegerPartitions[n];
%t Table[Count[g[n], p_ /; d[p] < f[p]], {n, 0, z}] (* A239954 *)
%t Table[Count[g[n], p_ /; d[p] <= f[p]], {n, 0, z}] (* A239955 *)
%t Table[Count[g[n], p_ /; d[p] == f[p]], {n, 0, z}] (* this sequence *)
%t Table[Count[g[n], p_ /; d[p] > f[p]], {n, 0, z}] (* A034296 *)
%t Table[Count[g[n], p_ /; d[p] >= f[p]], {n, 0, z}] (* A239958 *)
%o (PARI)
%o A_x(N) = {my(x='x+O('x^N), g = sum(m=1,N, sum(i=m+2,N, x^(i+m)/((1-x^i)*(1-x^m)) * sum(j=m+1,i-1, (1-x^j)/(x^j) * prod(k=m+1,i-1, (x^k/(1-x^k)))))));
%o concat([0,0,0,0],Vec(g))}
%o A_x(51) \\ _John Tyler Rascoe_, Mar 16 2024
%Y Cf. A239954, A239955, A034296, A239958.
%K nonn,easy
%O 0,7
%A _Clark Kimberling_, Mar 30 2014
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