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A367205
Number of nonnegative sequences of integers S with the properties that (1) sum(S) + length(S) = n and (2) there exists a nonnegative sequence whose Euler transform begins with S starting at index 1.
1
1, 2, 3, 6, 9, 16, 27, 45, 74, 125, 205, 343, 564, 934, 1535, 2536, 4165, 6855, 11249, 18465
OFFSET
1,2
COMMENTS
a(n) <= 2^(n-1), which is the number of nonnegative sequences S with sum(S) + length(S) = n.
The candidate sequences are related to the row n of A228369, by subtracting 1 from each term.
EXAMPLE
For n = 4 the a(4) = 6 sequences are
1) (0,0,0,0) because (1,0,0,0,0,...) = Euler(0,0,0,0,...),
2) (0,0,1) because (1,0,0,1,...) = Euler(0,0,1,...),
3) (0,1,0) because (1,0,1,0,...) = Euler(0,1,0,...),
4) (0,2) because (1,0,2,...) = Euler(0,2,...),
5) (1,1) because (1,1,1,...) = Euler(1,0,...) and
6) (3) because (1,3,...) = Euler(3,...).
The lexicographically earliest such sequences are:
1) A000007 = Euler(0,0,0,0,...)
2) A079978 = Euler(0,0,1,0,...)
3) A000035 = Euler(0,1,0,0,...)
4) A142150 = Euler(0,2,0,0,...)
5) A000012 = Euler(1,0,0,0,...)
6) A000217 = Euler(3,0,0,0,...)
Note that (2,0) and (1,0,0) are not the 1-indexed prefix of the Euler transform of a nonnegative sequence.
MATHEMATICA
A367205[n_] :=
Select[EulerInvTransform /@ (Map[# - 1 &, #] & /@
Join @@ Permutations /@ IntegerPartitions[n]),
AllTrue[#, # >= 0 &] &]
CROSSREFS
KEYWORD
nonn,more
AUTHOR
Peter Kagey, Nov 10 2023
STATUS
approved