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A357459
The total number of fixed points among all partitions of n, when parts are written in nondecreasing order.
1
0, 1, 1, 3, 4, 7, 10, 17, 22, 34, 46, 66, 88, 123, 160, 218, 283, 375, 482, 630, 799, 1030, 1299, 1651, 2066, 2602, 3230, 4032, 4976, 6157, 7554, 9288, 11326, 13837, 16793, 20393, 24632, 29763, 35783, 43031, 51527, 61683, 73577, 87729, 104252, 123834, 146664
OFFSET
0,4
COMMENTS
For instance, the partition (1,3,3,3,5) = (y(1),y(2),y(3),y(4),y(5)) has 3 fixed points, since y(i) = i for i=1,3,5.
LINKS
A. Blecher and A. Knopfmacher, Fixed points and matching points in partitions, Ramanujan J. 58 (2022), 23-41.
FORMULA
G.f.: (Product_{k>=1}(1/(1-q^k)))*Sum_{n>=1}q^(2*n-1)*Product_{k=n..2*n-2}(1-q^k).
EXAMPLE
The 7 partitions of 5 are (1,1,1,1,1), (1,1,1,2), (1,2,2), (1,1,3), (1,4), (2,3), and (5), containing 1, 1, 2, 2, 1, 0, and 0 fixed points, respectively, and so a(5) = 1+1+2+2+1+0+0=7.
CROSSREFS
Cf. A001522 (parts decreasing), A099036.
Sequence in context: A256912 A134591 A058611 * A098613 A261037 A280423
KEYWORD
nonn
AUTHOR
Jeremy Lovejoy, Sep 29 2022
STATUS
approved