OFFSET
0,2
COMMENTS
Number of rooted two-face n-edge maps in the plane (planar with a distinguished outside face). - Valery A. Liskovets, Mar 17 2005
Total number of returns to the x axis in all lattice paths using steps (1,1) and (1,-1) from the origin to (2n,0). Cf. A108747. - Geoffrey Critzer, Jan 30 2012
Total depth of all leaves in all binary trees on 2n+1 nodes. - Marko Riedel, Sep 10 2016
REFERENCES
H. W. Gould, Combinatorial Identities, Morgantown, WV, 1972. p. 32.
Hojoo Lee, Posting to Number Theory List, Feb 18 2002.
V. A. Liskovets and T. R. Walsh, Enumeration of unrooted maps on the plane, Rapport technique, UQAM, No. 2005-01, Montreal, Canada, 2005.
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..175
Dennis E. Davenport, Lara K. Pudwell, Louis W. Shapiro, and Leon C. Woodson, The Boundary of Ordered Trees, Journal of Integer Sequences, Vol. 18 (2015), Article 15.5.8.
Nicolle González, Pamela E. Harris, Gordon Rojas Kirby, Mariana Smit Vega Garcia, and Bridget Eileen Tenner, Pinnacle sets of signed permutations, arXiv:2301.02628 [math.CO], 2023.
Guo-Niu Han, Enumeration of Standard Puzzles, 2011. [Cached copy]
Guo-Niu Han, Enumeration of Standard Puzzles, arXiv:2006.14070 [math.CO], 2020.
Marko R. Riedel, Average depth of a leaf in a binary tree, Math.Stackexchange.com.
V. A. Liskovets and T. R. Walsh, Counting unrooted maps on the plane, Advances in Applied Math., 36(4) (2006), 364-387.
FORMULA
G.f.: 1/(1 - 4*x) - 1/sqrt(1 - 4*x) = C(x)*2*x/(1 - 4*x) where C(x) = g.f. for Catalan numbers A000108.
a(n) = Sum_{k >= 1} binomial(2*m-2*k, m-k) * binomial(2*k, k).
a(n+1) = 4*a(n) + 2*C(n), where C(n) = Catalan numbers.
a(n) = 2*A000346(n-1) for n > 0.
a(n) = A045621(2*n).
Conjecture: n*a(n) + 2*(3-4*n)*a(n-1) + 8*(2*n-3)*a(n-2) = 0. - R. J. Mathar, Apr 01 2012
Recurrence (an alternative): n*a(n) = 2^9*(2*n - 9)*a(n-5) + 2^8*(18 - 5*n)*a(n-4) + 2^6*(10*n - 27)*a(n-3) + 2^5*(9 - 5*n)*a(n-2) + 2*(10*n - 9)*a(n-1), n >= 5. - Fung Lam, Mar 22 2014
Asymptotics: a(n) ~ 2^(2*n)*(1 - 1/sqrt(n*Pi)). - Fung Lam, Mar 22 2014
E.g.f.: (exp(2*x) - BesselI(0, 2*x))*exp(2*x). - Ilya Gutkovskiy, Sep 10 2016
a(n) = (-1)^(n+1)*binomial(-n, n + 1)*hypergeom([1, 2*n + 1], [n + 2], 1/2). - Peter Luschny, Nov 29 2023
MAPLE
MATHEMATICA
nn=20; c=(1-(1-4x)^(1/2))/(2x); D[CoefficientList[ Series[ 1/(1-2y x c), {x, 0, nn}], x], y]/.y->1 (* Geoffrey Critzer, Jan 30 2012 *)
PROG
(PARI) a(n)=if(n<0, 0, 4^n-binomial(2*n, n))
(Magma) [4^n - Binomial(2*n, n): n in [0..35]]; // Vincenzo Librandi, Jun 07 2011
(PARI) x='x+O('x^100); concat(0, Vec(1/(1-4*x)-1/sqrt(1-4*x))) \\ Altug Alkan, Dec 29 2015
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
N. J. A. Sloane, Mar 23 2002
STATUS
approved