A182747 Bisection (odd part) of number of partitions that do not contain 1 as a part A002865.

0, 1, 2, 4, 8, 14, 24, 41, 66, 105, 165, 253, 383, 574, 847, 1238, 1794, 2573, 3660, 5170, 7245, 10087, 13959, 19196, 26252, 35717, 48342, 65121, 87331, 116600, 155038, 205343, 270928, 356169, 466610, 609237, 792906, 1028764, 1330772, 1716486, 2207851

Offset

0,3

Comments

a(n+1) = number of partitions p of 2n 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

Formula

a(n) = p(2*n+1)-p(2*n), where p is the partition function, A000041. - George Beck, Aug 14 2011

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+1, 2*n+1):
seq(a(n), n=0..40);  # Alois P. Heinz, Dec 01 2010

Mathematica

f[n_] := Table[PartitionsP[2 k + 1] - PartitionsP[2 k], {k, 0, n}] (* George Beck, Aug 14 2011 *)
(* also *)
Table[Count[IntegerPartitions[2 n], 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+1, 2*n+1]; Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Aug 29 2016, after Alois P. Heinz *)

Crossrefs

Cf. A000041, A002865, A058695, A135010, A138121, A182740, A182742, A182743, A182746.

Sequence in context: A164402 A164165 A164168 * A164406 A178982 A164397

Adjacent sequences: A182744 A182745 A182746 * A182748 A182749 A182750

Keyword

nonn,easy

Author

Omar E. Pol, Dec 01 2010

Extensions

More terms from Alois P. Heinz, Dec 01 2010

Status

approved