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Number of partitions of n into distinct parts >= 3.
10

%I #37 Nov 09 2023 12:20:02

%S 1,0,0,1,1,1,1,2,2,3,3,4,5,6,7,9,10,12,15,17,20,24,28,32,38,44,51,59,

%T 68,78,91,103,118,136,155,176,201,228,259,294,332,375,425,478,538,607,

%U 681,764,858,961,1075,1203,1343,1499,1673,1863,2073,2308,2564,2847,3161,3504

%N Number of partitions of n into distinct parts >= 3.

%H Alois P. Heinz, <a href="/A025148/b025148.txt">Table of n, a(n) for n = 0..1000</a>

%H 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 G.f.: Product_{k>=3} (1+x^k).

%F a(n) = A096749(n+2). - _R. J. Mathar_, Jul 31 2008

%F G.f.: sum(n>=0, x^(n*(n+5)/2) / prod(k=1..n, 1-x^k) ); special case of g.f. for partitions into distinct parts >= L, sum(n>=0, x^(n*(n+2*L-1)/2) / prod(k=1..n, 1-x^k) ). - _Joerg Arndt_, Mar 24 2011

%F G.f.: sum(n>=2, x^(n*(n+1)/2-3) / prod(k=1..n-2, 1-x^k) ), a special case of the g.f. for partitions into distinct parts >= L, sum(n>=L-1, x^(n*(n+1)/2-L*(L-1)/2) / prod(k=1..n-(L-1), 1-x^k) ). - _Joerg Arndt_, Mar 27 2011

%F a(n) + a(n+1) + a(n+2) + a(n+3) = A000009(n+3). - _Vaclav Kotesovec_, Oct 22 2015

%F a(n) ~ 1/4 * A000009(n). - _Vaclav Kotesovec_, Oct 22 2015

%p with(combstruct) ; sys := {L = PowerSet(Sequence(Z,card>2)) }; seq( count([L,sys],size=i), i=0..56 ); # _Zerinvary Lajos_, Mar 08 2007

%p A025148 := proc(n) mul(1+x^k,k=3..n+1) ; expand(%) ; coeftayl(%,x=0,n) ; end proc: # _R. J. Mathar_, Mar 28 2011

%p # third Maple program:

%p b:= proc(n, i) option remember;

%p `if`(n=0, 1, `if`((i-2)*(i+3)/2<n, 0,

%p add(b(n-i*j, i-1), j=0..min(1, n/i))))

%p end:

%p a:= n-> b(n$2):

%p seq(a(n), n=0..100); # _Alois P. Heinz_, Feb 07 2014

%t d[n_] := Select[IntegerPartitions[n], Max[Length /@ Split@#] == 1 && Min[#] >= 3 &]; Table[d[n], {n, 16}] (* strict partitions, parts >= 3 *)

%t Table[Length[d[n]], {n, 40}] (* A025148 for n >= 1 *)

%t (* _Clark Kimberling_, Mar 07 2014 *)

%t b[n_, i_] := b[n, i] = If[n==0, 1, If[(i-2)*(i+3)/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_ *)

%Y Cf. A025147.

%K nonn

%O 0,8

%A _Clark Kimberling_