OFFSET
1,3
REFERENCES
F. N. David, M. G. Kendall and D. E. Barton, Symmetric Function and Allied Tables, Cambridge, 1966, p. 262. (Terms for n>=13 are incorrect.)
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
LINKS
Alois P. Heinz, Table of n, a(n) for n = 1..450 (first 100 terms from Max Alekseyev)
Max A. Alekseyev, On the number of permutations with bounded runs length, arXiv preprint arXiv:1205.4581 [math.CO], 2012-2013. - From N. J. A. Sloane, Oct 23 2012
Richard Ehrenborg and JiYoon Jung, Descent pattern avoidance, arXiv preprint:1312.2027 [math.CO], 6 Dec 2013.
FORMULA
a(n) ~ c * d^n * n!, where d = 0.92403585760753647721113386869798700855648617941... is the root of the equation 8 - 2*sin(sqrt(phi)/d) * (2*sqrt(5*(phi-1)) * cosh(sqrt(phi-1)/d) + 2*sinh(sqrt(phi-1)/d)) + 2*cos(sqrt(phi)/d) * (6*cosh(sqrt(phi-1)/d) + 2*sqrt(5*phi) * sinh(sqrt(phi-1)/d)) = 0, phi = A001622 = (1+sqrt(5))/2 is the golden ratio and c = 1.259371257828351725264434486385284120241474052544197367866029465830756911... - Vaclav Kotesovec, Sep 06 2014, updated Aug 18 2018
MATHEMATICA
length = 3;
g[u_, o_, t_] := g[u, o, t] = If[u+o == 0, 1, Sum[g[o + j - 1, u - j, 2], {j, 1, u}] + If[t<length, Sum[g[u + j - 1, o - j, t+1], {j, 1, o}], 0]];
b[u_, o_, t_] := b[u, o, t] = If[t == length, g[u, o, t], Sum[b[o + j - 1, u - j, 2], {j, 1, u}] + Sum[b[u + j - 1, o - j, t + 1], {j, 1, o}]];
a[n_] := Sum[b[j - 1, n - j, 1], {j, 1, n}];
Table[a[n], {n, 1, 30}] (* Jean-Fran�ois Alcover, Aug 18 2018, after Alois P. Heinz *)
CROSSREFS
KEYWORD
nonn
AUTHOR
EXTENSIONS
Corrected and extended by Max Alekseyev at the suggestion of Sean A. Irvine, May 04 2012
STATUS
approved