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A005317
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a(n) = (2^n + C(2*n,n))/2.
(Formerly M1460)
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6
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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
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OFFSET
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0,2
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COMMENTS
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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
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REFERENCES
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N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
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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.
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FORMULA
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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
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MAPLE
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f := n->(2^n+binomial(2*n, n))/2;
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MATHEMATICA
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Table[(2^n + Binomial[2 n, n])/2, {n, 0, 26}] (* Michael De Vlieger, Feb 01 2016 *)
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PROG
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(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
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CROSSREFS
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Cf. A108958.
Sequence in context: A221586 A258312 A123020 * A126566 A112808 A088927
Adjacent sequences: A005314 A005315 A005316 * A005318 A005319 A005320
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KEYWORD
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nonn,easy
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AUTHOR
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N. J. A. Sloane and Peter Fishburn
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STATUS
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approved
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