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A374782 Number of partitions of n that do not have a fixed point that is also a fixed point of the conjugate partition.

%I A374782 #17 Jul 22 2024 15:42:06
%S A374782 1,0,2,3,4,5,8,11,17,23,33,43,60,77,104,134,177,226,295,373,480,604,
%T A374782 766,957,1204,1492,1860,2294,2836,3477,4273,5209,6362,7721,9375,11326,
%U A374782 13687,16460,19799,23720,28406,33901,40443,48092,57159,67747,80237,94799
%N A374782 Number of partitions of n that do not have a fixed point that is also a fixed point of the conjugate partition.
%H A374782 Alois P. Heinz, <a href="/A374782/b374782.txt">Table of n, a(n) for n = 0..2500</a>
%H A374782 Wikipedia, <a href="https://en.wikipedia.org/wiki/Partition_(number_theory)">Partition (number theory)</a>
%F A374782 a(n) = A000041(n) - A188674(n) for n > 0, a(0) = 1.
%e A374782 a(0) = 1: the empty partition.
%e A374782 a(2) = 2: 2, 11.
%e A374782 a(3) = 3: 3, 21, 111.
%e A374782 a(4) = 4: 4, 31, 211, 1111.
%e A374782 a(5) = 5: 5, 41, 311, 2111, 11111.
%e A374782 a(6) = 8: 6, 33, 51, 222, 411, 3111, 21111, 111111.
%e A374782 a(7) = 11: 7, 43, 61, 322, 331, 511, 2221, 4111, 31111, 211111, 1111111.
%p A374782 b:= proc(n, i, p) option remember; `if`(n=0, 1,
%p A374782      `if`(i<1, 0, b(n, i-1, p)+add(`if`(i=p+j, 0,
%p A374782       b(n-i*j, min(n-i*j, i-1), p+j)), j=1..n/i)))
%p A374782     end:
%p A374782 a:= n-> b(n$2, 0):
%p A374782 seq(a(n), n=0..47);
%Y A374782 Cf. A000041, A064428, A114701, A188674, A352828.
%K A374782 nonn
%O A374782 0,3
%A A374782 _Alois P. Heinz_, Jul 19 2024