OFFSET
0,3
COMMENTS
Number of rooted isthmusless planar maps with n+1 faces and 2 vertices. - Dan Drake, Aug 08 2005
a(n) = total area below all Dyck (n+1)-paths and above the lowest possible Dyck path, namely, UDUD...UD (taking upsteps of unit length). For example, the areas below the 5 Dyck 3-paths UUUDDD, UUDUDD, UDUUDD, UUDDUD, UDUDUD are 3,2,1,1,0 respectively, yielding a(2)=3+2+1+1+0=7. - David Callan, Jul 03 2006
REFERENCES
D. Phulara and L. W. Shapiro, Descendants in ordered trees with a marked vertex, Congressus Numerantium, 205 (2011), 121-128.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
LINKS
Seiichi Manyama, Table of n, a(n) for n = 0..1660
Jason Bandlow and Kendra Killpatrick, An area-to-inv bijection between Dyck paths and 312-avoiding permutations,Electron. J. Combin. 8 (2001), no. 1, Research Paper 40, 16 pp.
Miklós Bóna, Surprising Symmetries in Objects Counted by Catalan Numbers, Electronic J. Combin., 19 (2012), #P62, eq. (5).
Pudwell, Lara; Scholten, Connor; Schrock, Tyler; Serrato, Alexa Noncontiguous pattern containment in binary trees, ISRN Comb. 2014, Article ID 316535, 8 p. (2014), Table 1.
R. P. Stanley, F. Zanello, The Catalan case of Armstrong's conjecture on core partitions, arXiv preprint arXiv:1312.4352 [math.CO], 2013.
T. R. S. Walsh and A. B. Lehman, Counting rooted maps by genus. III: Nonseparable maps, J. Combinatorial Theory Ser. B 18 (1975), 222-259.
FORMULA
G.f.: c(x)^3*x/(1-4x) where c(x) = g.f. for the Catalan numbers A000108. - Philippe Deléham, Jun 02 2013
a(n) = Integral_{x=0..4} x^n*W(x)*dx, n >= 0, is the integral representation as n-th moment of a signed weight function W(x), where W(x) = W_a(x) + W_c(x), with W_a(x) = 2*Dirac(x-4), which is the discrete (atomic) part, and W_c(x) = (1/(2*Pi))*(1-x)*sqrt(x/(4-x)) is the continuous part of W(x): W_c(0) = W_c(1) = 0, W_c(x) > 0 for x < 1, lim_{x->4} W_c(x) = -oo. - Karol A. Penson, Jul 31 2013 [edited by Michel Marcus, Mar 14 2020]
(n+2)*a(n) + (-9*n-10)*a(n-1) + 2*(12*n+1)*a(n-2) + 8*(-2*n+3)*a(n-3) = 0. - R. J. Mathar, Mar 30 2014
a(n) = Sum_{k=0..n} binomial(2*(n+1), n-k-1). - Vladimir Kruchinin, Oct 23 2016
0 = a(n)*(+256*a(n+1) - 992*a(n+2) + 520*a(n+3) - 72*a(n+4)) + a(n+1)*(+224*a(n+1) + 344*a(n+2) - 398*a(n+3) + 70*a(n+4)) + a(n+2)*(+6*a(n+2) + 59*a(n+3) - 17*a(n+4)) + a(n+3)*(-a(n+3) + a(n+4)), for all n >= 0. - Michael Somos, Oct 23 2016
a(n) = [x^n] x/((1 - 2*x)*(1 - x)^(n+3)). - Ilya Gutkovskiy, Oct 25 2017
From Seiichi Manyama, Jul 29 2025: (Start)
a(n) = Sum_{k=0..n-1} binomial(2*k+1+l,k) * binomial(2*n-2*k-l,n-k-1) for every real number l.
a(n) = Sum_{k=0..n-1} 2^(n-k-1) * binomial(n+k+2,k). (End)
EXAMPLE
G.f. = x + 7*x^2 + 37*x^3 + 176*x^4 + 794*x^5 + 3473*x^6 + 14893*x^7 + 63004*x^8 + ...
MAPLE
f := n->2^(2*n+1)-binomial(2*n+3, n+1)+binomial(2*n+1, n); seq(f(n), n=0..30);
MATHEMATICA
Table[2^(2 n + 1) - Binomial[2 n + 3, n + 1] +
Binomial[2 n + 1, n], {n, 0, 30}] (* Wesley Ivan Hurt, Mar 30 2014 *)
PROG
(Maxima)
a(n):=sum(binomial(2*(n+1), n-k-1), k, 0, n); /* Vladimir Kruchinin, Oct 23 2016 */
CROSSREFS
KEYWORD
nonn
AUTHOR
STATUS
approved