login
A318980
Number of integer partitions of n whose parts plus 1 are relatively prime.
7
0, 0, 1, 1, 4, 5, 9, 13, 21, 29, 43, 56, 79, 109, 146, 192, 254, 329, 428, 553, 707, 900, 1139, 1434, 1800, 2251, 2799, 3472, 4286, 5275, 6469, 7918, 9655, 11755, 14252, 17248, 20817, 25084, 30134, 36142, 43235, 51644, 61548, 73241, 86961, 103108, 122010
OFFSET
1,5
LINKS
FORMULA
G.f.: Sum_{d>=1} mu(d)*(-1 + 1/(Prod_{k>=2/d} 1 - x^(k*d - 1))). - Andrew Howroyd, Oct 17 2019
EXAMPLE
The a(7) = 9 partitions are (61), (43), (421), (4111), (322), (3211), (2221), (22111), (211111).
The a(8) = 13 partitions:
(62),
(332), (422), (431), (521), (611),
(3221), (4211),
(22211), (32111), (41111),
(221111),
(2111111).
MATHEMATICA
Table[Length[Select[IntegerPartitions[n], GCD@@(#+1)==1&]], {n, 30}]
PROG
(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
KEYWORD
nonn
AUTHOR
Gus Wiseman, Sep 06 2018
STATUS
approved