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

1, 4, 7, 11, 15, 19, 25, 30, 37, 44, 53, 61, 72, 82, 95, 107, 122, 136, 154, 170, 190, 209, 232, 253, 279, 303, 332, 359, 391, 421, 457, 490, 529, 566, 609, 649, 696, 740, 791, 839, 894, 946, 1006, 1062, 1126, 1187, 1256, 1321, 1395, 1465, 1544, 1619, 1703

Offset: 14

Author

Alois P. Heinz, Jun 23 2014

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 14..1000
• N. Guru Sharan and Armin Straub, Partitions with Durfee triangles of fixed size, arXiv:2507.19047 [math.CO], 2025. See p. 5.

Crossrefs

Column k=4 of A238406.

Programs

PARI

p_q(k) = {prod(j=1, k, 1-q^j); }
GB_q(N, M)= {p_q(N+M)/(p_q(M)*p_q(N)); }
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))));
Vec(g)}
A_q(70) \\ John Tyler Rascoe, Apr 23 2024

Formula

Conjectures from Chai Wah Wu, Apr 18 2024: (Start)

a(n) = a(n-1) + a(n-2) - 2*a(n-5) + a(n-8) + a(n-9) - a(n-10) for n > 26.

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)

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