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A029760 A sum with next-to-central binomial coefficients of even order, Catalan related. 10
1, 8, 47, 244, 1186, 5536, 25147, 112028, 491870, 2135440, 9188406, 39249768, 166656772, 704069248, 2961699667, 12412521388, 51854046982, 216013684528, 897632738722, 3721813363288, 15401045060572, 63616796642368, 262357557683422, 1080387930269464 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
COMMENTS
Proof by induction.
a(n) = total area below paths consisting of steps east (1,0) and north (0,1) from (0,0) to (n+2,n+2) that stay weakly below y=x. For example, the two paths with n=0 are
. _|.....|
_|.....__|
The first has area 1 below it, the second area 0 and so a(0)=1. - David Callan, Dec 09 2004
Convolution of A000346 with A001700. - Philippe Deléham, May 19 2009
LINKS
Ran Pan, Jeffrey B. Remmel, Paired patterns in lattice paths, arXiv:1601.07988 [math.CO], 2016.
Sittipong Thamrongpairoj, Dowling Set Partitions, and Positional Marked Patterns, Ph. D. Dissertation, University of California-San Diego (2019).
FORMULA
a(n) = 4^(n+1)*Sum_{k=1..n+1} binomial(2k, k-1)/4^k = ((n+3)^2)*C(n+2)/2-2^(2*n+3), C = Catalan. Also a(n+1)=4*a(n)+binomial(2(n+2), n+1).
G.f.: (d/dx)c(x)/(1-4*x), where c(x) = g.f. for Catalan numbers; convolution of A001791 and powers of 4. G.f. also c(x)^2/(1-4*x)^(3/2); convolution of Catalan numbers A000108 C(n), n >= 1, with A002457; convolution of A008549(n), n >= 1, with A000984 (central binomial coefficients).
a(n) = Sum_{k=0..n+1} A039598(n+1,k)*k^2. - Philippe Deléham, Dec 16 2007
MATHEMATICA
a[n_] := (n+3)^2 CatalanNumber[n+2]/2 - 2^(2n+3);
Table[a[n], {n, 0, 23}] (* Jean-François Alcover, Sep 25 2018 *)
CROSSREFS
Sequence in context: A099110 A106393 A300167 * A139262 A026900 A016198
KEYWORD
nonn
AUTHOR
STATUS
approved

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Last modified March 19 03:21 EDT 2024. Contains 370952 sequences. (Running on oeis4.)