cp's OEIS Frontend

A026831 Number of partitions of n into distinct parts, the least being 10.

%I A026831 #23 Nov 25 2020 07:04:03
%S A026831 0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,
%T A026831 2,3,3,4,4,5,5,6,6,7,7,8,9,10,11,13,14,16,18,20,22,25,27,30,33,36,40,
%U A026831 44,48,53,59,64,71,78,86,94,104,113,125,136,149,163,179,194,213,232,254,276,302
%N A026831 Number of partitions of n into distinct parts, the least being 10.
%H A026831 Alois P. Heinz, <a href="/A026831/b026831.txt">Table of n, a(n) for n = 0..1000</a>
%F A026831 a(n) = A096740(n-10), n>10. - _R. J. Mathar_, Jul 31 2008
%F A026831 G.f.: x^10*Product_{j>=11} (1+x^j). - _R. J. Mathar_, Jul 31 2008
%F A026831 G.f.: Sum_{k>=1} x^(k*(k + 19)/2) / Product_{j=1..k-1} (1 - x^j). - _Ilya Gutkovskiy_, Nov 25 2020
%e A026831 Say n = 11. There is no way to partition 11 into n distinct parts if one of the least parts is 10 since 11 = 10 + x where x >= 10 has no solutions. Hence a(11) = 0.
%p A026831 b:= proc(n, i) option remember;
%p A026831       `if`(n=0, 1, `if`((i-10)*(i+11)/2<n, 0,
%p A026831        add(b(n-i*j, i-1), j=0..min(1, n/i))))
%p A026831     end:
%p A026831 a:= n-> `if`(n<10, 0, b(n-10$2)):
%p A026831 seq(a(n), n=0..100); # _Alois P. Heinz_, Feb 07 2014
%t A026831 b[n_, i_] := b[n, i] = If[n == 0, 1, If[(i-10)*(i+11)/2 < n, 0, Sum[b[n-i*j, i-1], {j, 0, Min[1, n/i]}]]]; a[n_] := If[n<10, 0, b[n-10, n-10]]; Table[a[n], {n, 0, 100}] (* _Jean-François Alcover_, Jun 24 2015, after _Alois P. Heinz_ *)
%t A026831 Join[{0}, Table[Count[Last /@ Select[IntegerPartitions@n, DeleteDuplicates[#] == # &], 10], {n, 66}]] (* _Robert Price_, Jun 13 2020 *)
%Y A026831 Cf. A025147, A025155, A096740.
%K A026831 nonn
%O A026831 0,34
%A A026831 _Clark Kimberling_