cp's OEIS Frontend

A244240 Number of partitions of n into 4 parts such that every i-th smallest part (counted with multiplicity) is different from i.

%I A244240 #17 Jul 28 2025 09:00:13
%S A244240 1,4,7,11,15,19,25,30,37,44,53,61,72,82,95,107,122,136,154,170,190,
%T A244240 209,232,253,279,303,332,359,391,421,457,490,529,566,609,649,696,740,
%U A244240 791,839,894,946,1006,1062,1126,1187,1256,1321,1395,1465,1544,1619,1703
%N A244240 Number of partitions of n into 4 parts such that every i-th smallest part (counted with multiplicity) is different from i.
%H A244240 Alois P. Heinz, <a href="/A244240/b244240.txt">Table of n, a(n) for n = 14..1000</a>
%H A244240 N. Guru Sharan and Armin Straub, <a href="https://arxiv.org/abs/2507.19047">Partitions with Durfee triangles of fixed size</a>, arXiv:2507.19047 [math.CO], 2025. See p. 5.
%F A244240 Conjectures from _Chai Wah Wu_, Apr 18 2024: (Start)
%F A244240 a(n) = a(n-1) + a(n-2) - 2*a(n-5) + a(n-8) + a(n-9) - a(n-10) for n > 26.
%F A244240 G.f.: x^14*(-x^4 + x + 1)*(x^8 - x^5 - 2*x^4 + 2*x + 1)/((x - 1)^4*(x + 1)^2*(x^2 + 1)*(x^2 + x + 1)). (End)
%F A244240 G.f.: Sum_{i>2} q^(8+i) * ( q_binomial(3,i) - q^2 - q^3 - Sum_{j=0..i} (q^j) ). - _John Tyler Rascoe_, Apr 23 2024
%o A244240 (PARI)
%o A244240 p_q(k) = {prod(j=1, k, 1-q^j); }
%o A244240 GB_q(N, M)= {p_q(N+M)/(p_q(M)*p_q(N)); }
%o A244240 A_q(N) = {my(q='q+O('q^N), g=sum(i=3,N, q^(8+i) * (GB_q(3,i) - q^2 - q^3 - sum(j=0,i, q^j))));
%o A244240 Vec(g)}
%o A244240 A_q(70) \\ _John Tyler Rascoe_, Apr 23 2024
%Y A244240 Column k=4 of A238406.
%K A244240 nonn
%O A244240 14,2
%A A244240 _Alois P. Heinz_, Jun 23 2014