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A239951 Number of partitions of n such that (number of distinct parts) > least part. 5
0, 0, 0, 1, 2, 4, 6, 10, 14, 22, 30, 44, 59, 84, 109, 151, 195, 261, 335, 440, 558, 723, 909, 1160, 1452, 1829, 2272, 2839, 3503, 4336, 5326, 6542, 7984, 9756, 11842, 14376, 17382, 20985, 25255, 30355, 36372, 43528, 51960, 61925, 73645, 87460, 103648, 122650 (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) + A239949(n) = A000041(n) for n >= 0.

EXAMPLE

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

MAPLE

b:= proc(n, i, d) option remember; `if`(n=0, 1, `if`(i<=d, 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..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, 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, 80}] (* Jean-Fran├žois Alcover, Nov 17 2015, after Alois P. Heinz *)

CROSSREFS

Cf. A239948, A239949, A239950, A239952.

Sequence in context: A103445 A001747 A048670 * A077625 A027383 A280611

Adjacent sequences:  A239948 A239949 A239950 * A239952 A239953 A239954

KEYWORD

nonn,easy

AUTHOR

Clark Kimberling, Mar 30 2014

STATUS

approved

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Last modified February 18 03:43 EST 2018. Contains 299298 sequences. (Running on oeis4.)