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 A000259 Number of certain rooted planar maps. (Formerly M2943 N1185) 2
 1, 3, 13, 63, 326, 1761, 9808, 55895, 324301, 1908878, 11369744, 68395917, 414927215, 2535523154, 15592255913, 96419104103, 599176447614, 3739845108057, 23435007764606, 147374772979438, 929790132901804, 5883377105975922, 37328490926964481, 237427707464042693 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS a(n) is also the number of North-East lattice paths from (0,0) to (2n,n) that begin with a North step, end with an East step, and do not bounce off the line y =1/2 x. Similarly, also the number of paths that begin with an East step, end with a North step, and do not bounce off the line y=1/2 x. - Michael D. Weiner, Aug 03 2017 REFERENCES N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence). N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence). LINKS Michael De Vlieger, Table of n, a(n) for n = 1..1211 D. Birmajer, J. Gil, and M. Weiner, Bounce statistics for rational lattice paths, arXiv:1707.09918 [math.CO], 2017. W. G. Brown, Enumeration of non-separable planar maps, Canad. J. Math., 15 (1963), 526-545. W. G. Brown, Enumeration of non-separable planar maps [Annotated scanned copy] FORMULA a(n) = Sum_{k = 1..n} (-1)^(k-1)*C(3n, n-k)*k/n*F(k-2) where F(k) is the k-th Fibonacci number (A000045) and F(-1) = 1. - Michael D. Weiner, Aug 03 2017 EXAMPLE For n = 2 the a(2) = 3 is counting the following three paths EEEENN, EEENEN, ENEEEN. The path EENEEN is excluded as it bounces off the line y = (1/2) x at the point (2, 1). - Michael D. Weiner, Aug 03 2017 MAPLE with(linalg): T := proc(n, k) if k<=n then k*add((2*j-k)*(j-1)!*(3*n-j-k-1)!/(j-k)!/(j-k)!/(2*k-j)!/(n-j)!, j=k..min(n, 2*k))/(2*n-k)! else 0 fi end: A := matrix(30, 30, T): seq(add(A[i, j], j=1..i), i=1..30); # Emeric Deutsch, Mar 03 2004 R := RootOf(x-t*(t-1)^2, t); ogf := series(1/((1-R-R^2)*(R-1)^2), x=0, 30); # Mark van Hoeij, Nov 08 2011 MATHEMATICA R = Root[#^3-2#^2+#-x&, 1]; CoefficientList[1/((1-R-R^2)*(R-1)^2) + O[x]^30, x] (* Jean-François Alcover, Feb 06 2016, after Mark van Hoeij *) Table[Sum[(-1)^(k - 1)*Binomial[3 n, n - k]*k/n*Fibonacci[k - 2], {k, n}], {n, 21}] (* Michael De Vlieger, Aug 04 2017 *) PROG (PARI) a(n) = sum(k = 1, n, (-1)^(k-1)*binomial(3*n, n-k)*k/n*fibonacci(k-2)); \\ Michel Marcus, Aug 04 2017 (MAGMA) [&+[(-1)^(k-1)*Binomial(3*n, n-k)*k/n*Fibonacci(k - 2):k in [0..n]]: n in [1..30]]; // Vincenzo Librandi, Aug 05 2017 (Python) from sympy import binomial, fibonacci def a(n): return sum([(-1)**(k - 1)*binomial(3*n, n - k)*k/n*fibonacci(k - 2) for k in xrange(1, n + 1)]) print map(a, xrange(1, 31)) # Indranil Ghosh, Aug 05 2017 CROSSREFS Row sums of A046651. Sequence in context: A001850 A130525 A243280 * A007855 A193112 A192729 Adjacent sequences:  A000256 A000257 A000258 * A000260 A000261 A000262 KEYWORD nonn AUTHOR EXTENSIONS More terms from Emeric Deutsch, Mar 03 2004 STATUS approved

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Last modified August 18 00:28 EDT 2018. Contains 313817 sequences. (Running on oeis4.)