A210249 Number of partitions of n in which all parts are less than n/2.

1, 0, 0, 1, 1, 3, 4, 8, 10, 18, 23, 37, 47, 71, 90, 131, 164, 230, 288, 393, 488, 653, 807, 1060, 1303, 1686, 2063, 2637, 3210, 4057, 4920, 6158, 7434, 9228, 11098, 13671, 16380, 20040, 23928, 29098, 34624, 41869, 49668, 59755, 70667, 84626, 99795, 118991

Offset: 0

Author

L. Edson Jeffery, Mar 19 2012

Keywords

• nonn

Comments

Also, a(n) gives the number of partitions of 2*n in which all parts are even and less than n.

Also, number of nonpalindromic partitions of n, n >= 1. In other words: a(n) is the number of partitions of n into parts which cannot be listed in palindromic order, n >= 1. - Omar E. Pol, Jan 11 2014

Examples

			a(7) = 8, because 3+3+1 = 3+2+2 = 3+2+1+1 = 3+1+1+1+1 = 2+2+2+1 = 2+2+1+1+1 = 2+1+1+1+1+1 = 1+1+1+1+1+1+1, exhausting the partitions of the indicated class for n=7.
		

Links

• David A. Corneth, Table of n, a(n) for n = 0..10000 (first 1001 terms from Alois P. Heinz)

Crossrefs

Row sums of triangle A124278, for n >= 3.

Cf. A000041, A025065.

Programs

Maple

b:= proc(n, i) option remember;
      if n=0 then 1
    elif i<1 then 0
    else b(n, i-1) +`if`(i>n, 0, b(n-i, i))
      fi
    end:
a:= n-> b(n, ceil(n/2)-1):
seq (a(n), n=0..50);  # Alois P. Heinz, Mar 19 2012

Mathematica

b[n_, i_] := b[n, i] = If[n==0, 1, If[i<1, 0, b[n, i-1] + If[i>n, 0, b[n-i, i]]]]; a[n_] := b[n, Ceiling[n/2]-1]; Table[a[n], {n, 0, 50}] (* Jean-François Alcover, Jan 09 2016, after Alois P. Heinz *)
Table[Count[IntegerPartitions[n],?(Max[#]<n/2&)],{n,0,50}] (* _Harvey P. Dale, Aug 12 2025 *)

Formula

a(n) = A000041(n) - A025065(n), n >= 1. - Omar E. Pol, Jan 11 2014

Extensions

More terms from Alois P. Heinz, Mar 19 2012