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%I #43 Jun 20 2022 04:40:41
%S 1,1,2,4,7,12,21,34,55,88,137,210,320,478,708,1039,1507,2167,3094,
%T 4378,6153,8591,11914,16424,22519,30701,41646,56224,75547,101066,
%U 134647,178651,236131,310962,408046,533623,695578,903811,1170827,1512301,1947826,2501928
%N Bisection (even part) of number of partitions that do not contain 1 as a part A002865.
%C a(n+1) is the number of partitions p of 2n-1 such that (number of parts of p) is a part of p, for n >=0. - _Clark Kimberling_, Mar 02 2014
%H Alois P. Heinz, <a href="/A182746/b182746.txt">Table of n, a(n) for n = 0..1000</a>
%H Marco Baggio, Vasilis Niarchos, Kyriakos Papadodimas, and Gideon Vos, <a href="http://arxiv.org/abs/1610.07612">Large-N correlation functions in N = 2 superconformal QCD</a>, arXiv preprint arXiv:1610.07612 [hep-th], 2016.
%H K. Blum, <a href="https://arxiv.org/abs/2103.03196">Bounds on the Number of Graphical Partitions</a>, arXiv:2103.03196 [math.CO], 2021. See Table on p. 7.
%F a(n) = p(2*n) - p(2*n-1), where p is the partition function, A000041. - _George Beck_, Jun 05 2017 [Shifted by _Georg Fischer_, Jun 20 2022]
%p b:= proc(n, i) option remember;
%p if n<0 then 0
%p elif n=0 then 1
%p elif i<2 then 0
%p else b(n, i-1) +b(n-i, i)
%p fi
%p end:
%p a:= n-> b(2*n, 2*n):
%p seq(a(n), n=0..40); # _Alois P. Heinz_, Dec 01 2010
%t Table[Count[IntegerPartitions[2 n -1], p_ /; MemberQ[p, Length[p]]], {n, 20}] (* _Clark Kimberling_, Mar 02 2014 *)
%t b[n_, i_] := b[n, i] = Which[n<0, 0, n==0, 1, i<2, 0, True, b[n, i-1] + b[n-i, i]]; a[n_] := b[2*n, 2*n]; Table[a[n], {n, 0, 40}] (* _Jean-François Alcover_, Sep 21 2015, after _Alois P. Heinz_ *)
%t a[n_] := PartitionsP[2*n] - PartitionsP[2*n - 1]; Table[a[n], {n, 0, 40}] (* _George Beck_, Jun 05 2017 *)
%o (PARI) a(n)=numbpart(2*n)-numbpart(2*n-1) \\ _Charles R Greathouse IV_, Jun 06 2017
%Y Cf. A000041, A002865, A058696, A135010, A138121, A182740, A182742, A182743, A182747.
%K nonn,easy
%O 0,3
%A _Omar E. Pol_, Dec 01 2010
%E More terms from _Alois P. Heinz_, Dec 01 2010
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