A210249 - OEIS

%I #33 Jun 19 2024 15:09:17

%S 1,0,0,1,1,3,4,8,10,18,23,37,47,71,90,131,164,230,288,393,488,653,807,

%T 1060,1303,1686,2063,2637,3210,4057,4920,6158,7434,9228,11098,13671,

%U 16380,20040,23928,29098,34624,41869,49668,59755,70667,84626,99795,118991

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

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

%C 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

%H David A. Corneth, <a href="/A210249/b210249.txt">Table of n, a(n) for n = 0..10000</a> (first 1001 terms from Alois P. Heinz)

%F a(n) = A000041(n) - A025065(n), n >= 1. - _Omar E. Pol_, Jan 11 2014

%e 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.

%p b:= proc(n, i) option remember;

%p       if n=0 then 1

%p     elif i<1 then 0

%p     else b(n, i-1) +`if`(i>n, 0, b(n-i, i))

%p       fi

%p     end:

%p a:= n-> b(n, ceil(n/2)-1):

%p seq (a(n), n=0..50);  # _Alois P. Heinz_, Mar 19 2012

%t 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_ *)

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

%Y Cf. A000041, A025065.

%K nonn

%O 0,6

%A _L. Edson Jeffery_, Mar 19 2012

%E More terms from _Alois P. Heinz_, Mar 19 2012