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

%I A210249 #35 Aug 12 2025 12:17:57
%S A210249 1,0,0,1,1,3,4,8,10,18,23,37,47,71,90,131,164,230,288,393,488,653,807,
%T A210249 1060,1303,1686,2063,2637,3210,4057,4920,6158,7434,9228,11098,13671,
%U A210249 16380,20040,23928,29098,34624,41869,49668,59755,70667,84626,99795,118991
%N A210249 Number of partitions of n in which all parts are less than n/2.
%C A210249 Also, a(n) gives the number of partitions of 2*n in which all parts are even and less than n.
%C A210249 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 A210249 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 A210249 a(n) = A000041(n) - A025065(n), n >= 1. - _Omar E. Pol_, Jan 11 2014
%e A210249 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 A210249 b:= proc(n, i) option remember;
%p A210249       if n=0 then 1
%p A210249     elif i<1 then 0
%p A210249     else b(n, i-1) +`if`(i>n, 0, b(n-i, i))
%p A210249       fi
%p A210249     end:
%p A210249 a:= n-> b(n, ceil(n/2)-1):
%p A210249 seq (a(n), n=0..50);  # _Alois P. Heinz_, Mar 19 2012
%t A210249 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_ *)
%t A210249 Table[Count[IntegerPartitions[n],_?(Max[#]<n/2&)],{n,0,50}] (* _Harvey P. Dale_, Aug 12 2025 *)
%Y A210249 Row sums of triangle A124278, for n >= 3.
%Y A210249 Cf. A000041, A025065.
%K A210249 nonn
%O A210249 0,6
%A A210249 _L. Edson Jeffery_, Mar 19 2012
%E A210249 More terms from _Alois P. Heinz_, Mar 19 2012