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A318980 Number of integer partitions of n whose parts plus 1 are relatively prime.

%I A318980 #9 Oct 17 2019 14:45:28
%S A318980 0,0,1,1,4,5,9,13,21,29,43,56,79,109,146,192,254,329,428,553,707,900,
%T A318980 1139,1434,1800,2251,2799,3472,4286,5275,6469,7918,9655,11755,14252,
%U A318980 17248,20817,25084,30134,36142,43235,51644,61548,73241,86961,103108,122010
%N A318980 Number of integer partitions of n whose parts plus 1 are relatively prime.
%H A318980 Andrew Howroyd, <a href="/A318980/b318980.txt">Table of n, a(n) for n = 1..1000</a>
%F A318980 G.f.: Sum_{d>=1} mu(d)*(-1 + 1/(Prod_{k>=2/d} 1 - x^(k*d - 1))). - _Andrew Howroyd_, Oct 17 2019
%e A318980 The a(7) = 9 partitions are (61), (43), (421), (4111), (322), (3211), (2221), (22111), (211111).
%e A318980 The a(8) = 13 partitions:
%e A318980   (62),
%e A318980   (332), (422), (431), (521), (611),
%e A318980   (3221), (4211),
%e A318980   (22211), (32111), (41111),
%e A318980   (221111),
%e A318980   (2111111).
%t A318980 Table[Length[Select[IntegerPartitions[n],GCD@@(#+1)==1&]],{n,30}]
%o A318980 (PARI) seq(n)={Vec(sum(d=1, n+1, moebius(d)*(-1 + 1/prod(k=ceil(2/d), (n+1)\d, 1 - x^(k*d-1) + O(x*x^n)))), -n)} \\ _Andrew Howroyd_, Oct 17 2019
%Y A318980 Cf. A000837, A007359, A018783, A051424, A281116, A289508, A289509, A318978.
%K A318980 nonn
%O A318980 1,5
%A A318980 _Gus Wiseman_, Sep 06 2018