cp's OEIS Frontend

A380481 Number of partitions of n into distinct parts less than n and not a multiple of 3.

%I A380481 #53 Jul 26 2025 10:51:18
%S A380481 1,0,0,1,0,1,2,2,2,3,3,4,6,6,7,9,9,11,14,15,17,20,22,25,30,33,37,42,
%T A380481 46,52,60,66,73,82,90,101,114,125,138,153,168,186,207,227,249,274,300,
%U A380481 330,364,398,435,476,519,568,622,678,738,804,874,952,1038,1127,1223,1327,1438,1561,1694,1834,1984,2146,2320,2509,2714,2930,3161
%N A380481 Number of partitions of n into distinct parts less than n and not a multiple of 3.
%F A380481 G.f.: x^3/(1-x^3) - x/(1-x) + Product_{n>=0} (1+x^(3n+1))*(1+x^(3n+2)).
%F A380481 a(n) ~ exp(Pi*sqrt(2*n)/3) / (2^(5/4) * sqrt(3) * n^(3/4)). - _Vaclav Kotesovec_, Jul 25 2025
%F A380481 a(n) = A003105(n) - A011655(n). - _Alois P. Heinz_, Jul 25 2025
%e A380481 a(3) = 1: [2,1].
%e A380481 a(5) = 1: [4,1].
%e A380481 a(6) = 2: [5,1], [4,2].
%e A380481 a(7) = 2: [5,2], [4,2,1].
%e A380481 a(8) = 2: [7,1], [5,2,1].
%e A380481 a(9) = 3: [8,1], [7,2], [5,4].
%e A380481 a(10) = 3: [8,2], [7,2,1], [5,4,1].
%e A380481 a(11) = 4: [10,1], [8,2,1], [7,4], [5,4,2].
%p A380481 b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
%p A380481       b(n, i-1)+`if`(irem(i, 3)=0, 0, b(n-i, min(n-i, i-1)))))
%p A380481     end:
%p A380481 a:= n-> b(n, n-1):
%p A380481 seq(a(n), n=0..74);  # _Alois P. Heinz_, Jul 24 2025
%t A380481 CoefficientList[
%t A380481  Series[-q/QPochhammer[q, q, 1] + q^3/QPochhammer[q^3, q, 1] +
%t A380481     QPochhammer[-q, q^3]*QPochhammer[-q^2, q^3], {q, 0, 500}], q]
%Y A380481 Cf. A003105, A011655, A357457.
%K A380481 nonn
%O A380481 0,7
%A A380481 _Harman Kaur_, Jun 23 2025