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A026829 Number of partitions of n into distinct parts, the least being 8.

%I A026829 #23 Nov 25 2020 03:11:06
%S A026829 0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,3,3,4,4,5,
%T A026829 5,6,6,7,8,9,10,12,13,15,17,19,21,24,26,29,33,36,40,45,50,55,62,68,76,
%U A026829 84,93,102,114,125,138,152,168,184,204,223,246,270,297,325,358,391,429,470
%N A026829 Number of partitions of n into distinct parts, the least being 8.
%H A026829 Alois P. Heinz, <a href="/A026829/b026829.txt">Table of n, a(n) for n = 0..1000</a>
%F A026829 a(n) = A025154(n-8), n>8. - _R. J. Mathar_, Jul 31 2008
%F A026829 G.f.: x^8*Product_{j>=9} (1+x^j). - _R. J. Mathar_, Jul 31 2008
%F A026829 G.f.: Sum_{k>=1} x^(k*(k + 15)/2) / Product_{j=1..k-1} (1 - x^j). - _Ilya Gutkovskiy_, Nov 24 2020
%p A026829 b:= proc(n, i) option remember;
%p A026829       `if`(n=0, 1, `if`((i-8)*(i+9)/2<n, 0,
%p A026829        add(b(n-i*j, i-1), j=0..min(1, n/i))))
%p A026829     end:
%p A026829 a:= n-> `if`(n<8, 0, b(n-8$2)):
%p A026829 seq(a(n), n=0..100);  # _Alois P. Heinz_, Feb 07 2014
%t A026829 b[n_, i_] := b[n, i] = If[n == 0, 1, If[(i-8)*(i+9)/2 < n, 0, Sum[b[n - i*j, i - 1], {j, 0, Min[1, n/i]}]]]; a[n_] := If[n < 8, 0, b[n-8, n-8]]; Table[a[n], {n, 0, 100}] (* _Jean-François Alcover_, Jun 24 2015, after _Alois P. Heinz_ *)
%t A026829 Join[{0}, Table[Count[Last /@ Select[IntegerPartitions@n, DeleteDuplicates[#] == # &], 8], {n, 66}]] (* _Robert Price_, Jun 13 2020 *)
%Y A026829 Cf. A025147, A025154.
%K A026829 nonn
%O A026829 0,28
%A A026829 _Clark Kimberling_