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

0,3

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.

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]

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 *)

Prog

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

Crossrefs

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

Sequence in context: A178937 A168368 A305106 * A100482 A301762 A003293

Adjacent sequences: A182743 A182744 A182745 * A182747 A182748 A182749

Keyword

nonn,easy

Author

Omar E. Pol, Dec 01 2010

Extensions

More terms from Alois P. Heinz, Dec 01 2010

Status

approved