cp's OEIS Frontend

A229816 Number of partitions of n such that if the length is k then k is not a part.

%I A229816 #44 Aug 11 2025 10:50:23
%S A229816 1,0,2,2,4,5,9,11,18,23,34,44,63,80,111,142,190,242,319,402,522,655,
%T A229816 837,1045,1322,1638,2053,2532,3144,3857,4757,5803,7111,8636,10516,
%U A229816 12716,15404,18543,22355,26807,32168,38430,45929,54670,65088,77220,91599,108330,128077,151006,177974
%N A229816 Number of partitions of n such that if the length is k then k is not a part.
%C A229816 For example with n=5 neither 32 or 311 are allowed.
%C A229816 Conjecture: Also, for n>=1, a(n-1) is the total number of distinct parts of each partition of 2n with partition rank n. - _George Beck_, Jun 23 2019
%H A229816 Alois P. Heinz, <a href="/A229816/b229816.txt">Table of n, a(n) for n = 0..1000</a>
%H A229816 Atul Dixit, Gaurav Kumar, and Aviral Srivastava, <a href="https://arxiv.org/abs/2508.04359">Non-Rascoe partitions and a rank parity function associated to the Rogers-Ramanujan partitions</a>, arXiv:2508.04359 [math.CO], 2025. See references.
%F A229816 From _Joerg Arndt_, Sep 30 2013: (Start)
%F A229816 a(n) = A000041(n) - A002865(n-1), n>=1.
%F A229816 G.f.: 1/E(x) - x*(1-x)/E(x) where E(x) = Product_{k>=1} 1-x^k. (End)
%e A229816 a(2) = 2 : 2, 11.
%e A229816 a(6) = 9 : 6, 51, 411, 33, 3111, 222, 2211, 21111, 111111.
%p A229816 b:= proc(n, i, t) option remember; `if`(n=0, 1, `if`(i<t, 0,
%p A229816        b(n, i-1, t)+`if`(i>n, 0, b(n-i, i, t))))
%p A229816     end:
%p A229816 a:= n-> b(n$2, 1)-b((n-1)$2, 2):
%p A229816 seq(a(n), n=0..60);  # _Alois P. Heinz_, Sep 30 2013
%t A229816 nn=50;CoefficientList[Series[ Product[1/(1-x^i),{i,1,nn}]-x Product[1/(1-x^i),{i,2,nn}],{x,0,nn}],x] (* _Geoffrey Critzer_, Sep 30 2013 *)
%t A229816 Table[PartitionsP[n] - (PartitionsP[n - 1] - PartitionsP[n - 2]), {n, 0, 60}] (* _Vincenzo Librandi_, Jun 30 2019 *)
%o A229816 (PARI)
%o A229816 N=66;  x='x+O('x^N);
%o A229816 gf = 1/eta(x) - x*(1-x)/eta(x);
%o A229816 Vec( gf )
%o A229816 \\ _Joerg Arndt_, Sep 30 2013
%Y A229816 Cf. A116645.
%Y A229816 Cf. A002865 (partitions where the number of parts is itself a part).
%Y A229816 Cf. A000041, A002865.
%K A229816 nonn
%O A229816 0,3
%A A229816 _Jon Perry_, Sep 30 2013
%E A229816 Corrected a(8) and extended beyond a(9), _Joerg Arndt_, Sep 30 2013