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A000344
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a(n) = 5*binomial(2n, n-2)/(n+3).
(Formerly M3904 N1602)
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34
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1, 5, 20, 75, 275, 1001, 3640, 13260, 48450, 177650, 653752, 2414425, 8947575, 33266625, 124062000, 463991880, 1739969550, 6541168950, 24647883000, 93078189750, 352207870014, 1335293573130, 5071418015120, 19293438101000, 73514652074500, 280531912316292
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OFFSET
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2,2
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COMMENTS
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a(n-3) is the number of n-th generation vertices in the tree of sequences with unit increase labeled by 4 (cf. Zoran Sunic reference). - Benoit Cloitre, Oct 07 2003
Number of lattice paths from (0,0) to (n,n) with steps E=(1,0) and N=(0,1) which touch but do not cross the line x-y=2. Example: For n=3 there are the 5 paths EENENN, EENNEN, EENNNE, ENEENN, NEEENN. - Herbert Kociemba, May 24 2004
Number of standard tableaux of shape (n+2,n-2). - Emeric Deutsch, May 30 2004
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REFERENCES
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C. Krishnamachary and M. Bheemasena Rao, Determinants whose elements are Eulerian, prepared Bernoullian and other numbers, J. Indian Math. Soc., Vol. 14 (1922), pp. 55-62, 122-138 and 143-146.
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).
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LINKS
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Muniru A Asiru, Table of n, a(n) for n = 2..300(Terms 2..170 from Vincenzo Librandi)
Anwar Al Ghabra, K. Gopala Krishna, Patrick Labelle, and Vasilisa Shramchenko, Enumeration of multi-rooted plane trees, arXiv:2301.09765 [math.CO], 2023. (Cites this sequence as "A00344")
Jean-Luc Baril and Helmut Prodinger, Enumeration of partial Lukasiewicz paths, arXiv:2205.01383 [math.CO], 2022.
Paul Barry, A Catalan Transform and Related Transformations on Integer Sequences, Journal of Integer Sequences, Vol. 8 (2005), Article 05.4.5.
Dennis E. Davenport, Lara K. Pudwell, Louis W. Shapiro, and Leon C. Woodson, The Boundary of Ordered Trees, Journal of Integer Sequences, Vol. 18 (2015), Article 15.5.8.
Hilmar Haukur Gudmundsson, Dyck paths, standard Young tableaux, and pattern avoiding permutations, PU. M. A., Vol. 21, No. 2 (2010), pp. 265-284 (see Theorem 4.2 p. 275).
Richard K. Guy, Catwalks, sandsteps and Pascal pyramids, J. Integer Sequences, Vol. 3 (2000), Article #00.1.6.
V. E. Hoggatt, Jr. and M. Bicknell, Catalan and related sequences arising from inverses of Pascal's triangle matrices, Fib. Quart., Vol. 14, No. 5 (1976), pp. 395-405.
C. Krishnamachary and M. Bheemasena Rao, Determinants whose elements are Eulerian, prepared Bernoullian and other numbers, J. Indian Math. Soc., Vol. 14 (1922), pp. 55-62, 122-138 and 143-146. [Annotated scanned copy]
Athanasios Papoulis, A new method of inversion of the Laplace transform, Quart. Appl. Math., Vol. 14 (1957), pp. 405-414. [Annotated scan of selected pages]
Athanasios Papoulis, A new method of inversion of the Laplace transform, Quart. Applied Math., Vol. 14 (1956), pp. 405-414.
John Riordan, Letter to N. J. A. Sloane, Nov 10 1970.
John Riordan, The distribution of crossings of chords joining pairs of 2n points on a circle, Math. Comp., Vol. 29, No. 129 (1975), pp. 215-222.
Zoran Sunic, Self-Describing Sequences and the Catalan Family Tree, Electronic Journal of Combinatorics, Vol. 10 (2003) Article N5.
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FORMULA
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Integral representation as n-th moment of a function on [0, 4], in Maple notation: a(n)=int(x^n*((1/2)/Pi*x^(3/2)*(x^2-3*x+1)*(4-x)^(1/2)), x=0..4), n=0, 1..., for which offset=0. - Karol A. Penson, Oct 11 2001
Expansion of x^2*C^5, where C = (1-(1-4*x)^(1/2))/(2*x) is g.f. for Catalan numbers (A000108). - Herbert Kociemba, May 02 2004
Let A be the Toeplitz matrix of order n defined by: A[i,i-1]=-1, A[i,j]=Catalan(j-i), (i<=j), and A[i,j]=0, otherwise. Then, for n>=4, a(n-2)=(-1)^(n-4)*coeff(charpoly(A,x),x^4). - Milan Janjic, Jul 08 2010
a(n) = A000108(n+2) - 3*A000108(n+1)+ A000108(n). - David Scambler, May 20 2012
D-finite with recurrence: (n+3)*(n-2)*a(n) = 2*n*(2n-1)*a(n-1). - R. J. Mathar, Jun 27 2012
a(n) = A214292(2*n-1,n-3) for n > 2. - Reinhard Zumkeller, Jul 12 2012
0 = a(n)*(-528*a(n+1) + 9162*a(n+2) - 9295*a(n+3) + 1859*a(n+4)) + a(n+1)*(-1650*a(n+1) - 762*a(n+2) + 4188*a(n+3) - 946*a(n+4)) + a(n+2)*(-1050*a(n+2) - 126*a(n+3) + 84*a(n+4)) for all n in Z. - Michael Somos, May 28 2014
0 = a(n)*(a(n)*(+16*a(n+1) + 6*a(n+2)) + a(n+1)*(+66*a(n+1) - 105*a(n+2) + 40*a(n+3)) + a(n+2)*(-69*a(n+2) + 15*a(n+3))) +a(n+1)*(a(n+1)*(50*a(n+1) + 42*a(n+2) - 28*a(n+3)) +a(n+2)*(+12*a(n+2))) for all n in Z. - Michael Somos, May 28 2014
0 = a(n)^2*(-16*a(n+1)^2 - 38*a(n+1)*a(n+2) - 12*a(n+2)^2) + a(n)*a(n+1)*(-66*a(n+1)^2 + 149*a(n+1)*a(n+2) - 23*a(n+2)^2) + a(n+1)^2*(-50*a(n+1)^2 + 2*a(n+2)^2) for all n in Z. - Michael Somos, May 28 2014
From Ilya Gutkovskiy, Jan 22 2017: (Start)
E.g.f.: (x*(2 + x) * BesselI(0, 2*x) - (2+x+x^2) * BesselI(1, 2*x)) * exp(2*x)/x^2.
a(n) ~ 5*4^n/(sqrt(Pi)*n^(3/2)). (End)
a(n) = (1/(n+1))*Sum_{i=0..n-2} (-1)^(n+i)*(n-i+1)*binomial(2n+2,i), n >= 2. - Taras Goy, Aug 09 2018
G.f.: x^2* 2F1(5/2,3;6;4*x) . - R. J. Mathar, Jan 27 2020
From Amiram Eldar, Jan 02 2022: (Start)
Sum_{n>=2} 1/a(n) = 14/5 - 38*Pi/(45*sqrt(3)).
Sum_{n>=2} (-1)^n/a(n) = 1956*log(phi)/(125*sqrt(5)) - 316/125, where phi is the golden ratio (A001622). (End)
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EXAMPLE
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G.f. = x^2 + 5*x^3 + 20*x^4 + 75*x^5 + 275*x^6 + 1001*x^7 + 3640*x^8 + ...
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MAPLE
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A000344List := proc(m) local A, P, n; A := [1]; P := [1, 1, 1, 1];
for n from 1 to m - 2 do P := ListTools:-PartialSums([op(P), P[-1]]);
A := [op(A), P[-1]] od; A end: A000344List(27); # Peter Luschny, Mar 26 2022
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MATHEMATICA
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Table[5 Binomial[2n, n-2]/(n+3), {n, 2, 40}] (* or *) CoefficientList[Series[ (1-Sqrt[1-4 x]+x (-5+3 Sqrt[1-4 x]-(-5+Sqrt[1-4 x]) x))/(2 x^5), {x, 0, 38}], x] (* Harvey P. Dale, May 01 2011 *)
a[ n_] := If[ n < 0, 0, 5 Binomial[2 n, n - 2] / (n + 3)]; (* Michael Somos, May 28 2014 *)
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PROG
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(Magma) [5*Binomial(2*n, n-2)/(n+3): n in [2..30]]; // Vincenzo Librandi, May 03 2011
(PARI) a(n)=5*binomial(2*n, n-2)/(n+3) \\ Charles R Greathouse IV, Jul 25 2011
(GAP) List([2..30], n->5*Binomial(2*n, n-2)/(n+3)); # Muniru A Asiru, Aug 09 2018
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CROSSREFS
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T(n, n+5) for n=0, 1, 2, ..., array T as in A047072.
A diagonal of any of the essentially equivalent arrays A009766, A030237, A033184, A059365, A099039, A106566, A130020, A047072.
Cf. A000108, A000245, A002057, A003517, A000588, A003518, A003519, A001392, A001622.
Sequence in context: A030191 A093131 A224422 * A290922 A275909 A275908
Adjacent sequences: A000341 A000342 A000343 * A000345 A000346 A000347
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KEYWORD
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nonn,easy
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AUTHOR
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N. J. A. Sloane
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STATUS
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approved
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