login
Number of partitions of n into distinct parts >= 4.
6

%I #33 Nov 09 2023 12:19:56

%S 1,0,0,0,1,1,1,1,1,2,2,3,3,4,4,6,6,8,9,11,12,15,17,20,23,27,31,36,41,

%T 47,55,62,71,81,93,105,120,135,154,174,197,221,251,281,317,356,400,

%U 447,502,561,628,701,782,871,972,1081,1202,1336,1483,1645,1825,2021,2237,2476,2736

%N Number of partitions of n into distinct parts >= 4.

%H Alois P. Heinz, <a href="/A025149/b025149.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 a(n) = A026824(n+3). - _R. J. Mathar_, Jul 31 2008

%F G.f.: Product_{j>=4} (1+x^j). - _R. J. Mathar_, Jul 31 2008

%F G.f.: sum(n>=0, x^(n*(n+7)/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 From _Vaclav Kotesovec_, Oct 22 2015: (Start)

%F a(n) + a(n+1) + a(n+2) + 2*a(n+3) + a(n+4) + a(n+5) + a(n+6) = A000009(n+6).

%F a(n) ~ 1/8 * A000009(n).

%F (End)

%p g:=product(1+x^(j), j=4..54): gser:=series(g, x=0, 55): seq(coeff(gser, x, n), n=0..53); # _Zerinvary Lajos_, Mar 09 2007

%p # second Maple program:

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

%p `if`(n=0, 1, `if`((i-3)*(i+4)/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[#] >= 4 &]; Table[d[n], {n, 12}] (* strict partitions, parts >= 4 *)

%t Table[Length[d[n]], {n, 40}] (* A025149 for n >= 1 *)

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

%t b[n_, i_] := b[n, i] = If[n==0, 1, If[(i-3)*(i+4)/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 0,10

%A _Clark Kimberling_

%E More terms from _N. J. A. Sloane_, Sep 28 2008