A026376 a(n) is the number of integer strings s(0),...,s(n) counted by array T in A026374 that have s(n)=2; also a(n) = T(2n,n-1).

1, 6, 30, 144, 685, 3258, 15533, 74280, 356283, 1713690, 8263596, 39938616, 193419915, 938430990, 4560542550, 22195961280, 108171753355, 527816696850, 2578310320610, 12607504827600, 61706212037295, 302275142049870, 1481908332595625, 7270432009471224

Offset: 1

Author

Clark Kimberling

Keywords

• nonn
• changed

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)

Equals A026378 (1, 4, 17, 75, ...) convolved with A007317 (1, 2, 5, 15, 51, ...).

Equals A081671 (1, 3, 11, 45, ...) convolved with A002212 (1, 3, 10, 36, 137, ...).

(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.

Crossrefs

Cf. A006318, A002212, A081671, A007317, A026378.

Programs

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 *)

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

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