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%I #37 Jun 30 2019 22:42:44
%S 1,0,2,2,4,5,9,11,18,23,34,44,63,80,111,142,190,242,319,402,522,655,
%T 837,1045,1322,1638,2053,2532,3144,3857,4757,5803,7111,8636,10516,
%U 12716,15404,18543,22355,26807,32168,38430,45929,54670,65088,77220,91599,108330,128077,151006,177974
%N Number of partitions of n such that if the length is k then k is not a part.
%C For example with n=5 neither 32 or 311 are allowed.
%C 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 Alois P. Heinz, <a href="/A229816/b229816.txt">Table of n, a(n) for n = 0..1000</a>
%F a(n) = A000041(n) - A002865(n-1), n>=1. [_Joerg Arndt_, Sep 30 2013]
%F G.f.: 1/E(x) - x*(1-x)/E(x) where E(x) = Product_{k>=1} 1-x^k. [_Joerg Arndt_, Sep 30 2013]
%e a(2) = 2 : 2, 11.
%e a(6) = 9 : 6, 51, 411, 33, 3111, 222, 2211, 21111, 111111.
%p b:= proc(n, i, t) option remember; `if`(n=0, 1, `if`(i<t, 0,
%p b(n, i-1, t)+`if`(i>n, 0, b(n-i, i, t))))
%p end:
%p a:= n-> b(n$2, 1)-b((n-1)$2, 2):
%p seq(a(n), n=0..60); # _Alois P. Heinz_, Sep 30 2013
%t 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 Table[PartitionsP[n] - (PartitionsP[n - 1] - PartitionsP[n - 2]), {n, 0, 60}] (* _Vincenzo Librandi_, Juan 30 2019 *)
%o (PARI)
%o N=66; x='x+O('x^N);
%o gf = 1/eta(x) - x*(1-x)/eta(x);
%o Vec( gf )
%o \\ _Joerg Arndt_, Sep 30 2013
%Y Cf. A116645.
%Y Cf. A002865 (partitions where the number of parts is itself a part).
%Y Cf. A000041, A002865.
%K nonn
%O 0,3
%A _Jon Perry_, Sep 30 2013
%E Corrected a(8) and extended beyond a(9), _Joerg Arndt_, Sep 30 2013
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