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 A239949 Number of partitions of n such that (number of distinct parts) <= least part. 5

%I

%S 1,1,2,2,3,3,5,5,8,8,12,12,18,17,26,25,36,36,50,50,69,69,93,95,123,

%T 129,164,171,215,229,278,300,365,387,468,507,595,652,760,830,966,1055,

%U 1214,1336,1530,1674,1910,2104,2380,2617,2953,3253,3656,4019,4504

%N Number of partitions of n such that (number of distinct parts) <= least part.

%H Alois P. Heinz, <a href="/A239949/b239949.txt">Table of n, a(n) for n = 0..1000</a>

%F a(n) + A239951(n) = A000041(n) for n >= 0.

%e a(8) counts these 8 partitions: 8, 62, 53, 44, 422, 332, 2222, 11111111.

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

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

%p end:

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

%p seq(a(n), n=0..80); # _Alois P. Heinz_, Apr 02 2014

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

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

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

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

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

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

%t 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_] := b[n, n, 0]; Table[a[n], {n, 0, 80}] (* _Jean-François Alcover_, Nov 17 2015, after _Alois P. Heinz_ *)

%Y Cf. A239948, A239950, A239951, A239952.

%K nonn,easy

%O 0,3

%A _Clark Kimberling_, Mar 30 2014

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Last modified August 8 19:29 EDT 2020. Contains 336298 sequences. (Running on oeis4.)