OFFSET
0,7
LINKS
John Tyler Rascoe, Table of n, a(n) for n = 0..100
FORMULA
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
EXAMPLE
a(8) counts these 6 partitions: 53, 431, 422, 4211, 3311, 311111.
MATHEMATICA
z = 60; d[p_] := d[p] = Length[DeleteDuplicates[p]]; f[p_] := f[p] = Max[p] - Min[p]; g[n_] := g[n] = IntegerPartitions[n];
Table[Count[g[n], p_ /; d[p] < f[p]], {n, 0, z}] (* A239954 *)
Table[Count[g[n], p_ /; d[p] <= f[p]], {n, 0, z}] (* A239955 *)
Table[Count[g[n], p_ /; d[p] == f[p]], {n, 0, z}] (* this sequence *)
Table[Count[g[n], p_ /; d[p] > f[p]], {n, 0, z}] (* A034296 *)
Table[Count[g[n], p_ /; d[p] >= f[p]], {n, 0, z}] (* A239958 *)
PROG
(PARI)
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)))))));
concat([0, 0, 0, 0], Vec(g))}
A_x(51) \\ John Tyler Rascoe, Mar 16 2024
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Clark Kimberling, Mar 30 2014
STATUS
approved