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, 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.

Links

Table of n, a(n) for n=1..20.

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

Cf. A000007, A000012, A000035, A000217, A079978, A142150

Cf. A228369.

Sequence in context: A118033 A048810 A331680 * A327475 A017915 A114702

Adjacent sequences: A367202 A367203 A367204 * A367206 A367207 A367208

Keyword

nonn,more

Author

Peter Kagey, Nov 10 2023

Status

approved