OFFSET
0,2
COMMENTS
Hankel transform is A008619. - Paul Barry, Nov 13 2007
a(n) is the number of lattice paths from (0,0) to (n,n) using E(1,0) and N(0,1) as steps that horizontally cross the diagonal y = x with even many times. For example, a(2) = 5 because there are 6 paths in total and only one of them horizontally crosses the diagonal with odd many times, namely, NEEN. - Ran Pan, Feb 01 2016
REFERENCES
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..1664 (first 179 terms from Vincenzo Librandi)
Jelena Đokic, A short note on the order of the double reduced 2-factor transfer digraph for rectangular grid graphs, arXiv:2308.04155 [math.CO], 2023.
Jelena Đokić, Olga Bodroža-Pantić, and Ksenija Doroslovački, A spanning union of cycles in rectangular grid graphs, thick grid cylinders and Moebius strips, Transactions on Combinatorics (2023) Art. 27132.
T. Kløve, Generating functions for the number of permutations with limited displacement, Electron. J. Combin., 16 (2009), #R104. - From N. J. A. Sloane, May 04 2011.
Peter Fishburn, Letter to N. J. A. Sloane, Mar 1987
Mircea Merca, A Note on Cosine Power Sums J. Integer Sequences, Vol. 15 (2012), Article 12.5.3.
Ran Pan and Jeffrey B. Remmel, Paired patterns in lattice paths, arXiv:1601.07988 [math.CO], 2016.
Simon Plouffe, Master's Thesis, copy at the arXiv site, arXiv:0911.4975 [math.NT], 2009.
FORMULA
From Simon Plouffe, Feb 18 2011: (Start)
G.f.: (1/2)*(-4*x+1+(-(4*x-1)*(2*x-1)^2)^(1/2))/(4*x-1)/(2*x-1).
Recurrence: 0 = (-24-28*n-8*n^2)*a(n+1) + (18+22*n+6*n^2)*a(n+2) + (-3-4*n-n^2)*a(n+3), a(0)=1, a(1)=2, a(2)=5. (End)
a(n) = Sum_{k=0..floor(n/2)} (-1)^k*C(2*n, n-2*k), n > 0. - Mircea Merca, Jun 20 2011
E.g.f.: (exp(2*x)*(1+BesselI(0,2*x))/2 = G(0)/2; G(k) = 1 + (k)!/(P-2*x*(2*k+1)*(P^2)/(2*x*(2*k+1)*P+(k+1)^2*k!/G(k+1))), where P:=((2*k)!)/(2^k)/((k)!) (continued fraction). - Sergei N. Gladkovskii, Dec 20 2011
a(n) = Sum_{r=0..n} k*(k+1)/2 where k=C(n,r). - J. M. Bergot, Sep 04 2013
a(n) ~ 2^(2*n-1)/sqrt(n*Pi). - Stefano Spezia, Apr 17 2024
MAPLE
f := n->(2^n+binomial(2*n, n))/2;
MATHEMATICA
Table[(2^n + Binomial[2 n, n])/2, {n, 0, 26}] (* Michael De Vlieger, Feb 01 2016 *)
PROG
(Magma) [(2^n+Binomial(2*n, n))/2: n in [0..26]]; // Bruno Berselli, Jun 20 2011
(Maxima) makelist(sum((-1)^k*binomial(2*n, n-2*k), k, 0, floor(n/2)), n, 0, 26); \\ Bruno Berselli, Jun 20 2011
(PARI) a(n)=(2^n+binomial(2*n, n))/2 \\ Charles R Greathouse IV, Dec 20 2011
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
N. J. A. Sloane and Peter Fishburn
STATUS
approved