OFFSET
1,2
COMMENTS
Number of Schroeder paths (i.e., consisting of steps U=(1,1), D=(1,-1) and H=(2,0) and never going below the x-axis) from (0,0) to (2n+2,0), with exactly one peak at an even level. E.g., a(2)=6 because we have UUDDH, HUUDD, UDUUDD, UUDDUD, UUDHD and UHUDD. - Emeric Deutsch, Dec 28 2003
Number of left steps in all skew Dyck paths of semilength n+1. A skew Dyck path is a path in the first quadrant which begins at the origin, ends on the x-axis, consists of steps U=(1,1)(up), D=(1,-1)(down) and L=(-1,-1)(left) so that up and left steps do not overlap. The length of the path is defined to be the number of its steps. Example: a(2)=6 because in the 10 (=A002212(3)) skew Dyck paths of semilength 3 ( namely UDUUDL, UUUDLD, UUDUDL, UUUDDL, UUUDLL and five Dyck paths that have no left steps) we have altogether 6 left steps. - Emeric Deutsch, Aug 05 2007
From Gary W. Adamson, May 17 2009: (Start)
(End)
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 1..200
Emeric Deutsch, Emanuele Munarini, and Simone Rinaldi, Skew Dyck paths, J. Stat. Plann. Infer. 140 (8) (2010) 2191-2203.
Ricardo Gómez Aíza, Trees with flowers: A catalog of integer partition and integer composition trees with their asymptotic analysis, arXiv:2402.16111 [math.CO], 2024. See p. 18.
Toufik Mansour and José Luis Ramírez, Enumeration of Fuss-skew paths, Ann. Math. Inform. (2022) Vol. 55, 125-136. See p. 129.
FORMULA
E.g.f.: exp(3x)*I_1(2x), where I_1 is Bessel function. - Michael Somos, Sep 09 2002
G.f.: (1 - 3*z - t)/(2*z*t) where t = sqrt(1-6*z+5*z^2). - Emeric Deutsch, May 25 2003
a(n) = [t^(n+1)](1+3t+t^2)^n. a := n -> Sum_{j=ceiling((n+1)/2)..n} 3^(2j-n-1)*binomial(n, j)*binomial(j, n+1-j). - Emeric Deutsch, Jan 30 2004
a(n) = Sum_{k=0..n} binomial(n, k)*binomial(2k, k+1). - Paul Barry, Sep 20 2004
a(n) = n*A002212(n). - Emeric Deutsch, Aug 05 2007
D-finite with recurrence (n+1)*a(n) - 9*n*a(n-1) + (23*n-27)*a(n-2) + 15*(-n+2)*a(n-3) = 0. - R. J. Mathar, Dec 02 2012
a(n) ~ 5^(n+1/2) / (2*sqrt(Pi*n)). - Vaclav Kotesovec, Feb 13 2014
a(n) = n*hypergeometric([1, 3/2, 1-n],[1, 3],-4). - Peter Luschny, Sep 16 2014
a(n) = GegenbauerC(n-1, -n, -3/2). - Peter Luschny, May 09 2016
MAPLE
a := n -> simplify(GegenbauerC(n-1, -n, -3/2)):
seq(a(n), n=1..24); # Peter Luschny, May 09 2016
MATHEMATICA
Rest[CoefficientList[Series[(1-3*x-Sqrt[1-6*x+5*x^2])/(2*x*Sqrt[1-6*x+5*x^2]), {x, 0, 20}], x]] (* Vaclav Kotesovec, Feb 13 2014 *)
PROG
(PARI) a(n)=if(n<0, 0, polcoeff((1+3*x+x^2)^n, n-1))
(Sage)
A026376 = lambda n : n*hypergeometric([1, 3/2, 1-n], [1, 3], -4)
[round(A026376(n).n(100)) for n in (1..24)] # Peter Luschny, Sep 16 2014
(Sage) # Recurrence:
def A026376():
x, y, n = 1, 1, 1
while True:
x, y = y, ((6*n + 3)*y - (5*n - 5)*x) / (n + 2)
yield n*x
n += 1
a = A026376()
[next(a) for i in (1..24)] # Peter Luschny, Sep 16 2014
CROSSREFS
KEYWORD
nonn
AUTHOR
STATUS
approved