A000344 a(n) = 5*binomial(2n, n-2)/(n+3).
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
Offset: 2
Examples
G.f. = x^2 + 5*x^3 + 20*x^4 + 75*x^5 + 275*x^6 + 1001*x^7 + 3640*x^8 + ...
References
- 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).
Links
- 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.
- Olivier Danvy, Summa Summarum: Moessner's Theorem without Dynamic Programming, arXiv:2412.03127 [cs.DM], 2024. See p. 31.
- 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.
Crossrefs
Programs
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GAP
List([2..30],n->5*Binomial(2*n,n-2)/(n+3)); # Muniru A Asiru, Aug 09 2018
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Magma
[5*Binomial(2*n,n-2)/(n+3): n in [2..30]]; // Vincenzo Librandi, May 03 2011
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Maple
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
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 *) -
PARI
a(n)=5*binomial(2*n,n-2)/(n+3) \\ Charles R Greathouse IV, Jul 25 2011
Formula
Integral representation as n-th moment of a function on [0, 4]: a(n) = Integral_{x=0..4} x^n*((1/2)/Pi*x^(3/2)*(x^2-3*x+1)*(4-x)^(1/2)) dx, n >= 0, 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
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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