cp's OEIS Frontend

A025151 Number of partitions of n into distinct parts >= 6.

%I A025151 #24 Nov 09 2023 12:15:42
%S A025151 1,0,0,0,0,0,1,1,1,1,1,1,1,2,2,3,3,4,4,5,5,7,7,9,10,12,13,16,17,20,23,
%T A025151 26,29,34,38,43,49,55,62,70,79,88,100,111,125,140,157,174,196,217,243,
%U A025151 270,301,333,372,411,457,506,561,619,687,757,837,924,1019,1122,1238,1361,1498
%N A025151 Number of partitions of n into distinct parts >= 6.
%H A025151 Alois P. Heinz, <a href="/A025151/b025151.txt">Table of n, a(n) for n = 0..1000</a>
%H A025151 Kevin Beanland and Hung Viet Chu, <a href="https://arxiv.org/abs/2311.01926">On Schreier-type Sets, Partitions, and Compositions</a>, arXiv:2311.01926 [math.CO], 2023.
%F A025151 a(n) = A026826(n+5). - _R. J. Mathar_, Jul 31 2008
%F A025151 G.f.: Product_{j>=6} (1+x^j). - _R. J. Mathar_, Jul 31 2008
%F A025151 G.f.: Sum_{k>=0} x^(k*(k + 11)/2) / Product_{j=1..k} (1 - x^j). - _Ilya Gutkovskiy_, Nov 24 2020
%p A025151 b:= proc(n, i) option remember;
%p A025151       `if`(n=0, 1, `if`((i-5)*(i+6)/2<n, 0,
%p A025151        add(b(n-i*j, i-1), j=0..min(1, n/i))))
%p A025151     end:
%p A025151 a:= n-> b(n$2):
%p A025151 seq(a(n), n=0..100);  # _Alois P. Heinz_, Feb 07 2014
%t A025151 d[n_] := Select[IntegerPartitions[n], Max[Length /@ Split@#] == 1 && Min[#] >= 6 &]; Table[d[n], {n, 16}] (* strict partitions, parts >= 6 *)
%t A025151 Table[Length[d[n]], {n, 40}] (* A025151 for n >= 1 *)
%t A025151 (* _Clark Kimberling_, Mar 07 2014 *)
%t A025151 b[n_, i_] := b[n, i] = If[n == 0, 1, If[(i - 5)(i + 6)/2 < n, 0, Sum[b[n - i j, i - 1], {j, 0, Min[1, n/i]}]]]; a[n_] := b[n, n]; Table[a[n], {n, 0, 100}] (* _Jean-François Alcover_, Aug 29 2016, after _Alois P. Heinz_ *)
%Y A025151 Cf. A025147.
%K A025151 nonn,easy
%O A025151 0,14
%A A025151 _Clark Kimberling_