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%I #40 Aug 18 2020 18:08:44
%S 0,0,0,1,0,0,0,1,1,1,1,1,2,2,3,3,4,4,6,6,8,9,11,12,15,17,20,23,27,31,
%T 36,41,47,55,62,71,81,93,105,120,135,154,174,197,221,251,281,317,356,
%U 400,447,502,561,628,701,782,871,972,1081,1202,1336,1483,1645,1825,2021,2237,2476
%N Number of partitions of n into distinct parts, the least being 3.
%C Also, number of partitions of n such that if k is the largest part, then k occurs exactly 3 times and each of the numbers 1,2,...,k-1 occur at least once (these are the conjugates of the partitions described in the definition). Example: a(14)=3 because we have [3,3,3,2,2,1],[3,3,3,2,1,1,1] and [2,2,2,1,1,1,1,1,1,1,1]. - _Emeric Deutsch_, Apr 17 2006
%C For n > 3, a(n) is the Euler transform of [0,0,0,1,1,1,1] joined with the period 2 sequence [0,1, ...]. - _Georg Fischer_, Aug 18 2020
%H Alois P. Heinz, <a href="/A026824/b026824.txt">Table of n, a(n) for n = 0..1000</a>
%F From _Emeric Deutsch_, Apr 17 2006: (Start)
%F G.f.: (x^3)*Product_{j=4..infinity} (1+x^j).
%F G.f.: Sum_{k=1..infinity} x^(k*(k+5)/2)/(Product_{j=1..k-1} (1-x^j)). (End)
%F a(n) = A025149(n-3), n>3. - _R. J. Mathar_, Jul 31 2008
%F a(n) ~ exp(Pi*sqrt(n/3)) / (32*3^(1/4)*n^(3/4)). - _Vaclav Kotesovec_, Oct 30 2015
%e a(14) = 3 because we have [11,3], [7,4,3] and [6,5,3].
%p g:=x^3*product(1+x^j,j=4..80): gser:=series(g,x=0,70): seq(coeff(gser,x,n),n=1..59); # _Emeric Deutsch_, Apr 17 2006
%p # second Maple program:
%p b:= proc(n, i) option remember;
%p `if`(n=0, 1, `if`((i-3)*(i+4)/2<n, 0,
%p add(b(n-i*j, i-1), j=0..min(1, n/i))))
%p end:
%p a:= n-> `if`(n<3, 0, b(n-3$2)):
%p seq(a(n), n=0..60); # _Alois P. Heinz_, Feb 07 2014
%t b[n_, i_] := b[n, i] = If[n == 0, 1, If[(i-3)(i+4)/2 < n, 0, Sum[b[n-i*j, i-1], {j, 0, Min[1, n/i]}]]]; a[n_] := If[n<3, 0, b[n-3, n-3]]; Table[a[n], {n, 0, 60}] (* _Jean-François Alcover_, May 13 2015, after _Alois P. Heinz_ *)
%t nmax = 100; CoefficientList[Series[x^3/((1+x)*(1+x^2)*(1+x^3)) * Product[1+x^k, {k, 1, nmax}], {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Oct 30 2015 *)
%t Join[{0}, Table[Count[Last /@ Select[IntegerPartitions@n, DeleteDuplicates[#] == # &], 3], {n, 1, 66}]] (* _Robert Price_, Jun 13 2020 *)
%Y Cf. A025147, A025149.
%Y Cf. A096765 (least=1), A096749 (2), A022825 (4), A022826 (5), A022827 (6), A022828 (7), A022829 (8), A022830 (9), A022831 (10).
%K nonn
%O 0,13
%A _Clark Kimberling_
%E More terms from _Emeric Deutsch_, Apr 17 2006
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