cp's OEIS Frontend

A025149 Number of partitions of n into distinct parts >= 4.

%I A025149 #33 Nov 09 2023 12:19:56
%S A025149 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 A025149 47,55,62,71,81,93,105,120,135,154,174,197,221,251,281,317,356,400,
%U A025149 447,502,561,628,701,782,871,972,1081,1202,1336,1483,1645,1825,2021,2237,2476,2736
%N A025149 Number of partitions of n into distinct parts >= 4.
%H A025149 Alois P. Heinz, <a href="/A025149/b025149.txt">Table of n, a(n) for n = 0..1000</a>
%H A025149 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 A025149 a(n) = A026824(n+3). - _R. J. Mathar_, Jul 31 2008
%F A025149 G.f.: Product_{j>=4} (1+x^j). - _R. J. Mathar_, Jul 31 2008
%F A025149 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 A025149 From _Vaclav Kotesovec_, Oct 22 2015: (Start)
%F A025149 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 A025149 a(n) ~ 1/8 * A000009(n).
%F A025149 (End)
%p A025149 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 A025149 # second Maple program:
%p A025149 b:= proc(n, i) option remember;
%p A025149       `if`(n=0, 1, `if`((i-3)*(i+4)/2<n, 0,
%p A025149        add(b(n-i*j, i-1), j=0..min(1, n/i))))
%p A025149     end:
%p A025149 a:= n-> b(n$2):
%p A025149 seq(a(n), n=0..100);  # _Alois P. Heinz_, Feb 07 2014
%t A025149 d[n_] := Select[IntegerPartitions[n], Max[Length /@ Split@#] == 1 && Min[#] >= 4 &]; Table[d[n], {n, 12}] (* strict partitions, parts >= 4 *)
%t A025149 Table[Length[d[n]], {n, 40}] (* A025149 for n >= 1 *)
%t A025149 (* _Clark Kimberling_, Mar 07 2014 *)
%t A025149 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 A025149 Cf. A025147.
%K A025149 nonn
%O A025149 0,10
%A A025149 _Clark Kimberling_
%E A025149 More terms from _N. J. A. Sloane_, Sep 28 2008