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A239948 Number of partitions of n such that (number of distinct parts) < least part. 8
1, 0, 1, 1, 2, 1, 3, 2, 4, 4, 6, 6, 9, 9, 12, 14, 17, 18, 25, 26, 32, 38, 43, 49, 62, 65, 78, 92, 103, 114, 142, 151, 175, 203, 229, 252, 302, 323, 378, 422, 477, 524, 619, 661, 758, 847, 958, 1038, 1204, 1297, 1485, 1626, 1829, 1989, 2285, 2459, 2770, 3035 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,5

LINKS

Alois P. Heinz, Table of n, a(n) for n = 0..1000

FORMULA

a(n) + A239952(n) = A000041(n) for n >= 0.

EXAMPLE

a(10) counts these 6 partitions:  [10], [7,3], [6,4], [5,5], [4,3,3], [2,2,2,2,2].

MAPLE

b:= proc(n, i, d) option remember; `if`(n=0, 1, `if`(i<=d+1, 0,

      add(b(n-i*j, i-1, d+`if`(j=0, 0, 1)), j=0..n/i)))

    end:

a:= n-> b(n$2, 0):

seq(a(n), n=0..80);  # Alois P. Heinz, Apr 02 2014

MATHEMATICA

z = 50; d[p_] := d[p] = Length[DeleteDuplicates[p]]; f[n_] := f[n] = IntegerPartitions[n];

Table[Count[f[n], p_ /; d[p] < Min[p]], {n, 0, z}]  (*A239948*)

Table[Count[f[n], p_ /; d[p] <= Min[p]], {n, 0, z}] (*A239949*)

Table[Count[f[n], p_ /; d[p] == Min[p]], {n, 0, z}] (*A239950*)

Table[Count[f[n], p_ /; d[p] > Min[p]], {n, 0, z}]  (*A239951*)

Table[Count[f[n], p_ /; d[p] >= Min[p]], {n, 0, z}] (*A239952*)

b[n_, i_, d_] := b[n, i, d] = If[n==0, 1, If[i <= d+1, 0, Sum[b[n-i*j, i-1, d + If[j==0, 0, 1]], {j, 0, n/i}]]]; a[n_] := b[n, n, 0]; Table[a[n], {n, 0, 80}] (* Jean-Fran├žois Alcover, Oct 12 2015, after Alois P. Heinz *)

CROSSREFS

Cf. A239949, A239950, A239951, A239952.

Sequence in context: A001687 A159072 A116928 * A034391 A239243 A206738

Adjacent sequences:  A239945 A239946 A239947 * A239949 A239950 A239951

KEYWORD

nonn,easy

AUTHOR

Clark Kimberling, Mar 30 2014

STATUS

approved

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Last modified August 8 11:31 EDT 2020. Contains 336298 sequences. (Running on oeis4.)