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A239952 Number of partitions of n such that (number of distinct parts) >= least part. 5
0, 1, 1, 2, 3, 6, 8, 13, 18, 26, 36, 50, 68, 92, 123, 162, 214, 279, 360, 464, 595, 754, 959, 1206, 1513, 1893, 2358, 2918, 3615, 4451, 5462, 6691, 8174, 9940, 12081, 14631, 17675, 21314, 25637, 30763, 36861, 44059, 52555, 62600, 74417, 88287, 104600, 123716 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,4

LINKS

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

FORMULA

a(n) + A239948(n) = A000041(n) for n >= 0.

EXAMPLE

a(6) counts these 8 partitions:  51, 42, 411, 321, 3111, 2211, 21111, 111111.

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-> combinat[numbpart](n) -b(n$2, 0):

seq(a(n), n=0..60);  # 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_] := PartitionsP[n] - b[n, n, 0]; Table[a[n], {n, 0, 60}] (* Jean-Fran├žois Alcover, Nov 17 2015, after Alois P. Heinz *)

CROSSREFS

Cf. A239948, A239949, A239950, A239951.

Sequence in context: A251260 A022943 A068491 * A240076 A266771 A295342

Adjacent sequences:  A239949 A239950 A239951 * A239953 A239954 A239955

KEYWORD

nonn,easy

AUTHOR

Clark Kimberling, Mar 30 2014

STATUS

approved

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Last modified September 30 03:19 EDT 2020. Contains 337432 sequences. (Running on oeis4.)