OFFSET
1,2
COMMENTS
Number of "up" steps in all Motzkin paths of length n+1. E.g. a(2)=3 because in the four Motzkin paths of length 3, HHH, HUD, UDH and UHD, where H=(1,0), U=(1,1), D=(1,-1), we have altogether three U steps. - Emeric Deutsch, Dec 26 2003
a(n-1) = A111808(n,n-2) for n>1. - Reinhard Zumkeller, Aug 17 2005
a(n) = number of paths in the half-plane x>=0, from (0,0) to (n+1,2), and consisting of steps U=(1,1), D=(1,-1) and H=(1,0). For example, for n=2, we have the 3 paths: UUH, HUU, UHU. - José Luis Ramírez Ramírez, Apr 19 2015
REFERENCES
L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 78.
LINKS
G. C. Greubel, Table of n, a(n) for n = 1..1000 (terms 1..200 from T. D. Noe)
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 pp. 21-22.
Mark Shattuck, Subword Patterns in Smooth Words, Enum. Comb. Appl. (2024) Vol. 4, No. 4, Art. No. S2R32. See p. 6.
Eric Weisstein's World of Mathematics, Trinomial Coefficient.
FORMULA
E.g.f.: exp(x)*(2*x*BesselI(1, 2*x)+(x-2)*BesselI(2, 2*x))/x. - Vladeta Jovovic, Aug 21 2003
G.f.: [1-2z-z^2-(1-z)q]/(2z^3q), where q=sqrt(1-2z-3z^2). - Emeric Deutsch, Dec 26 2003
a(n) = Sum_{k=0..n+1} C(n+1,k)*C(n-k+1,k+2). - Paul Barry, Sep 20 2004
D-finite with recurrence (n+3)*(n-1)*a(n) -(n+1)*(2n+1)*a(n-2)-3*n*(n+1)*a(n-2)=0. - R. J. Mathar, Dec 08 2011
a(n) = n*(n+1)*hypergeom([(1-n)/2, 1-n/2], [3], 4)/2. - Peter Luschny, Nov 23 2014
G.f.: z*M(z)^2/(1-z-2*z^2*M(z)), where M(z) is the g.f. of Motzkin paths. - José Luis Ramírez Ramírez, Apr 19 2015
a(n) = GegenbauerC(n-1, -n-1, -1/2). - Peter Luschny, May 09 2016
a(n) = Sum_{k>0} k * A055151(n+1,k). - Alois P. Heinz, Mar 29 2020
MAPLE
seq( add(binomial(i+1, k)*binomial(i-k+1, k+2), k=0..floor(i/2)), i=1..30 ); # Detlef Pauly (dettodet(AT)yahoo.de), Nov 09 2001
a := n -> simplify(GegenbauerC(n-1, -n-1, -1/2)):
seq(a(n), n=1..26); # Peter Luschny, May 09 2016
MATHEMATICA
Table[Sum[Binomial[i + 1, k]*Binomial[i - k + 1, k + 2], {k, 0, Floor[i/2]}], {i, 30}] (* Michael De Vlieger, Apr 20 2015 *)
Table[GegenbauerC[n - 1, -n - 1, -1/2], {n, 1, 50}] (* G. C. Greubel, Feb 28 2017 *)
PROG
(Sage)
a = lambda n: n*(n+1)*hypergeometric([(1-n)/2, 1-n/2], [3], 4)/2
[simplify(a(n)) for n in (1..26)] # Peter Luschny, Nov 23 2014
(PARI) for(n=1, 25, print1(sum(k=0, n+1, binomial(n+1, k)*binomial(n-k+1, k+2)), ", ")) \\ G. C. Greubel, Feb 28 2017
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
EXTENSIONS
More terms from James A. Sellers, Feb 05 2000
STATUS
approved