cp's OEIS Frontend

A025150 Number of partitions of n into distinct parts >= 5.

%I A025150 #30 Nov 09 2023 12:19:52
%S A025150 1,0,0,0,0,1,1,1,1,1,1,2,2,3,3,4,4,5,6,7,8,10,11,13,15,17,20,23,26,30,
%T A025150 35,39,45,51,58,66,75,84,96,108,122,137,155,173,195,219,245,274,307,
%U A025150 342,383,427,475,529,589,654,727,807,894,991,1098,1214,1343,1485,1638,1809,1995
%N A025150 Number of partitions of n into distinct parts >= 5.
%H A025150 Alois P. Heinz, <a href="/A025150/b025150.txt">Table of n, a(n) for n = 0..1000</a>
%H A025150 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 A025150 G.f.: Product_{j>=5} (1+x^j). - _Emeric Deutsch_, Apr 17 2006
%F A025150 a(n) = A026825(n+4). - _R. J. Mathar_, Jul 31 2008
%F A025150 a(n) ~ exp(Pi*sqrt(n/3)) / (64*3^(1/4)*n^(3/4)). - _Vaclav Kotesovec_, Oct 30 2015
%F A025150 G.f.: Sum_{k>=0} x^(k*(k + 9)/2) / Product_{j=1..k} (1 - x^j). - _Ilya Gutkovskiy_, Nov 24 2020
%e A025150 a(12) = 2 because we have [12] and [7,5].
%p A025150 g:=product(1+x^j,j=5..70)-1: gser:=series(g,x=0,60): seq(coeff(gser,x,n), n=1..53); # _Emeric Deutsch_, Apr 17 2006
%p A025150 # second Maple program:
%p A025150 b:= proc(n, i) option remember;
%p A025150       `if`(n=0, 1, `if`((i-4)*(i+5)/2<n, 0,
%p A025150        add(b(n-i*j, i-1), j=0..min(1, n/i))))
%p A025150     end:
%p A025150 a:= n-> b(n$2):
%p A025150 seq(a(n), n=0..100);  # _Alois P. Heinz_, Feb 07 2014
%t A025150 d[n_] := Select[IntegerPartitions[n], Max[Length /@ Split@#] == 1 && Min[#] >= 5 &]; Table[d[n], {n, 16}] (* strict partitions, parts >= 5 *)
%t A025150 Table[Length[d[n]], {n, 40}] (* A025150 for n >= 1 *)
%t A025150 (* _Clark Kimberling_, Mar 07 2014 *)
%t A025150 b[n_, i_] := b[n, i] = If[n==0, 1, If[(i-4)*(i+5)/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_, Oct 22 2015, after _Alois P. Heinz_ *)
%t A025150 nmax = 100; CoefficientList[Series[Product[1+x^k, {k, 1, nmax}] / ((1+x)*(1+x^2)*(1+x^3)*(1+x^4)), {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Oct 30 2015 *)
%Y A025150 Cf. A025147.
%K A025150 nonn
%O A025150 0,12
%A A025150 _Clark Kimberling_