OFFSET
0,4
COMMENTS
Or row sums of the compressed triangle in A293783.
Conjecture: all terms are positive integers.
From David A. Corneth (with participation of Vladimir Shevelev), Oct 24 2017: (Start)
Conjecture is true. Proof.
1) Let C={c_1..c_n} be a permutation of {1..n}, d(C) be alternating sum c_1 - c_2 + ... +(-1)^(n-1)*c_n. Then max_{C in S_n}d(C) = A008794(n+1). Indeed, if n = 2*m, then evidently the maximum is reached on a C={2*m,1,2*m-1,2,...,m+1,m}; if n=2*m - 1, then the maximum is reached on a C={2*m-1,1,2*m-2,2,...,m-1,m}. In both cases max_{C in S_n}d(C) = m^2 = A008794(n+1). The number of distinct reaches of the maximum is, evidently, floor(n/2)!*floor((n+1)/2)! which is also Avi Peretz's representation (2001) of A010551(n). So, A293857(n) >= A010551(n) and a(n)>=1.
2) Consider two cases: a) there are no C in S_n for which d(C) = k^2 < A008794(n+1). Then A293857(n) = A010551(n) and a(n) = 1; b) there is C for which d(C) = k^2 < A008794(n+1). Then, as in 1) to reach k^2 in case n=2*m consider all (n/2)! permutations of {c_1,c_3,...,c_n} and all (n/2)! permutations of {c_2, c_4, ... , c_(n+1)), or in case n = 2*m-1, all ((n+1)/2)! permutations of {c_1,c_3,...,c_(2*m-1)} and ((n-1)/2)! permutations of {c_2,c_4,...,c_(2m-2)}. So we again have A010551(n) distinct reaches. If the same k^2 could be reached by another permutation C_1 (other than above permutations of C), then we again obtain A010551 distinct reaches, etc. So, A293857(n) is always divisible by A010551(n). (End)
LINKS
Peter J. C. Moses, Table of n, a(n) for n = 0..250
MAPLE
b:= proc(p, m, s) option remember; (n-> `if`(n=0, `if`(issqr(s), 1, 0),
`if`(p>0, b(p-1, m, s+n), 0)+`if`(m>0, b(p, m-1, s-n), 0)))(p+m)
end:
a:= n-> (t-> b(n-t, t, 0))(iquo(n, 2)):
seq(a(n), n=0..40); # Alois P. Heinz, Sep 17 2020
MATHEMATICA
a293984=Table[
possibleSums=Range[1/2-(-1)^n/2-Floor[n/2]^2, Floor[(n+1)/2]^2];
filteredSums=Select[possibleSums, IntegerQ[Sqrt[#]]&];
positions=Map[Flatten[{#, Position[possibleSums, #, 1]-1}]&, filteredSums];
Total[Map[SeriesCoefficient[QBinomial[n, Floor[(n+1)/2], q], {q, 0, #[[2]]/2}]&, positions]], {n, 20}] (* Peter J. C. Moses, Nov 05 2017 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Vladimir Shevelev, Oct 21 2017
EXTENSIONS
a(13)-a(30) from David A. Corneth, Oct 21 2017; a(31)-a(38) from Peter J. C. Moses, Nov 02 2017
a(0)=1 prepended by Alois P. Heinz, Sep 17 2020
STATUS
approved