OFFSET
3,2
REFERENCES
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
LINKS
G. Kreweras, Les préordres totaux compatibles avec un ordre partiel, Math. Sci. Humaines No. 53 (1976), 5-30.
G. Kreweras, Les préordres totaux compatibles avec un ordre partiel, Math. Sci. Humaines No. 53 (1976), 5-30. (Annotated scanned copy)
FORMULA
From Peter Luschny, Mar 26 2026: (Start)
The connection with the zig-zag Eulerian numbers and the Euler numbers is given by the P-Eulerian polynomials Zn which Kyle Petersen has introduced in the Maple section of A205497:
a(n) = Sum_{k=0..n} (k - 1) * [x^k] Zn(n).
The connection with the elementary symmetric polynomials A050446 is:
a(n) = Sum_{k=0..n} k * Sum_{j=0..k} (-1)^j * binomial(n + 1, j) * A050446(n, k - j). (End)
MAPLE
# After Alois P. Heinz in A000111:
b := proc(u, o) option remember;
`if`(u + o = 0, 1, add(b(o - 1 + j, u - j), j = 1..u)) end:
a := n -> (n-2)*b(n-1, 1)/2: seq(a(n), n = 3..23); # Peter Luschny, Oct 27 2017
MATHEMATICA
b[u_, o_] := b[u, o] = If[u+o == 0, 1, Sum[b[o-1+j, u-j], {j, 1, u}]];
a[n_] := (n-2) b[n-1, 1]/2;
Array[a, 22, 3] (* Jean-François Alcover, Jun 01 2019, from Maple *)
CROSSREFS
KEYWORD
nonn
AUTHOR
EXTENSIONS
More terms from Sean A. Irvine, Mar 12 2017
STATUS
approved
