A005317 a(n) = (2^n + C(2*n,n))/2. (Formerly M1460)

1, 2, 5, 14, 43, 142, 494, 1780, 6563, 24566, 92890, 353740, 1354126, 5204396, 20066492, 77575144, 300572963, 1166868646, 4537698722, 17672894044, 68923788698, 269129985796, 1052051579012, 4116719558104, 16123810230158, 63205319996092, 247959300028484

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) = binomial(2*n,n) - A108958(n). - Ran Pan, Feb 01 2016

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

Cf. A008619, A108958.

Sequence in context: A221586 A258312 A123020 * A126566 A112808 A369158

Adjacent sequences: A005314 A005315 A005316 * A005318 A005319 A005320

Keyword

nonn,easy,changed

Author

N. J. A. Sloane and Peter Fishburn

Status

approved