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A005317 a(n) = (2^n + C(2*n,n))/2.
(Formerly M1460)
6
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 (list; graph; refs; listen; history; text; internal format)
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

Vincenzo Librandi, Table of n, a(n) for n = 0..178

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.

P. 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 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[k*(k+1)/2 {k=C(n,r) 0<=r<=n}]. - J. M. Bergot, Sep 04 2013

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

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. A108958.

Sequence in context: A221586 A258312 A123020 * A126566 A112808 A088927

Adjacent sequences:  A005314 A005315 A005316 * A005318 A005319 A005320

KEYWORD

nonn,easy

AUTHOR

N. J. A. Sloane and Peter Fishburn

STATUS

approved

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Last modified September 26 06:54 EDT 2017. Contains 292502 sequences.