A182746 Bisection (even part) of number of partitions that do not contain 1 as a part A002865.

1, 1, 2, 4, 7, 12, 21, 34, 55, 88, 137, 210, 320, 478, 708, 1039, 1507, 2167, 3094, 4378, 6153, 8591, 11914, 16424, 22519, 30701, 41646, 56224, 75547, 101066, 134647, 178651, 236131, 310962, 408046, 533623, 695578, 903811, 1170827, 1512301, 1947826, 2501928

Offset: 0

Author

Omar E. Pol, Dec 01 2010

Keywords

• nonn
• easy

Comments

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

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..1000
• Marco Baggio, Vasilis Niarchos, Kyriakos Papadodimas, and Gideon Vos, Large-N correlation functions in N = 2 superconformal QCD, arXiv preprint arXiv:1610.07612 [hep-th], 2016.
• K. Blum, Bounds on the Number of Graphical Partitions, arXiv:2103.03196 [math.CO], 2021. See Table on p. 7.

Crossrefs

Cf. A000041, A002865, A058696, A135010, A138121, A182740, A182742, A182743, A182747.

Programs

Maple

b:= proc(n, i) option remember;
      if n<0 then 0
    elif n=0 then 1
    elif i<2 then 0
    else b(n, i-1) +b(n-i, i)
      fi
    end:
a:= n-> b(2*n, 2*n):
seq(a(n), n=0..40);  # Alois P. Heinz, Dec 01 2010

Mathematica

Table[Count[IntegerPartitions[2 n -1], p_ /; MemberQ[p, Length[p]]], {n, 20}]   (* Clark Kimberling, Mar 02 2014 *)
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 *)
a[n_] := PartitionsP[2*n] - PartitionsP[2*n - 1]; Table[a[n], {n, 0, 40}] (* George Beck, Jun 05 2017 *)

PARI

a(n)=numbpart(2*n)-numbpart(2*n-1) \\ Charles R Greathouse IV, Jun 06 2017

Formula

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]

Extensions

More terms from Alois P. Heinz, Dec 01 2010