|
| |
|
|
A143017
|
|
Number of {2-1-3, 2'^e-31}-avoiding permutations of size n (see definition in the Elizalde paper).
|
|
2
|
|
|
|
1, 2, 4, 9, 22, 56, 147, 396, 1088, 3036, 8582, 24524, 70727, 205594, 601756, 1771937, 5245544, 15602496, 46606356, 139753120, 420520000, 1269361000, 3842722454, 11663928644, 35490451807, 108232655126, 330760284892
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
1,2
|
|
|
COMMENTS
|
a(n) is the number of Dyck paths of semilength n for which all non-terminal descents are of odd length. For example, a(3) = 4 counts all 5 Dyck paths of semilength 3 except UUDDUD and a(4) = 9 counts, among others, UUUDUDDD and UUDUDUDD but not UUDDUUDD. - David Callan, Nov 13 2021
|
|
|
LINKS
|
|
|
|
FORMULA
|
a(n) = (1/n)*Sum_{k=0..floor(n/2)} 2*binomial(n,2k)*binomial(n-k,k-1) + n*binomial(n,2k+1)*binomial(n-k,k)/(n-k).
G.f. G(x) satisfies x*G^3 + (4x-2)*G^2 + (4x-1)*G + x = 0.
Conjecture: -8*n*(n+1)*a(n) + 4*n*(2*n+5)*a(n-1) + 4*n*(n+7)*a(n-2) + 2*(70*n^2-395*n+564)*a(n-3) + 2*(25*n^2-143*n+222)*a(n-4) + 4*(49*n-228)*(n-5)*a(n-5) - 45*(n-5)*(n-6)*a(n-6) = 0. - R. J. Mathar, Mar 14 2014
Recurrence (of order 4): 4*n*(n+1)*(91*n^2 - 217*n + 102)*a(n) = 6*n*(182*n^3 - 525*n^2 + 365*n - 78)*a(n-1) - 4*(91*n^4 - 399*n^3 - 136*n^2 + 990*n - 450)*a(n-2) + 12*(n-3)*(182*n^3 - 525*n^2 + 92*n + 140)*a(n-3) - 5*(n-4)*(n-3)*(91*n^2 - 35*n - 24)*a(n-4). - Vaclav Kotesovec, Mar 20 2014
|
|
|
MAPLE
|
a:=proc(n) options operator, arrow: (sum(2*binomial(n, 2*k)*binomial(n-k, k-1)+n*binomial(n, 2*k+1)*binomial(n-k, k)/(n-k), k=0..floor((1/2)*n)))/n end proc: seq(a(n), n=1..27);
|
|
|
MATHEMATICA
|
Table[1/n*Sum[2*Binomial[n, 2k]*Binomial[n-k, k-1]+ n*Binomial[n, 2k+1] *Binomial[n-k, k]/(n-k), {k, 0, n-1}], {n, 1, 20}] (* Vaclav Kotesovec, Mar 20 2014 *)
|
|
|
CROSSREFS
|
|
|
|
KEYWORD
|
nonn
|
|
|
AUTHOR
|
|
|
|
STATUS
|
approved
|
| |
|
|
