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A003518 a(n) = 8*binomial(2*n+1,n-3)/(n+5).
(Formerly M4529)

%I M4529

%S 1,8,44,208,910,3808,15504,62016,245157,961400,3749460,14567280,

%T 56448210,218349120,843621600,3257112960,12570420330,48507033744,

%U 187187399448,722477682080,2789279908316,10772391370048,41620603020640

%N a(n) = 8*binomial(2*n+1,n-3)/(n+5).

%C a(n-6) is the number of n-th generation nodes in the tree of sequences with unit increase labeled by 7 (cf. _Zoran Sunic_ reference). - _Benoit Cloitre_, Oct 07 2003

%C Number of standard tableaux of shape (n+4,n-3). - _Emeric Deutsch_, May 30 2004

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%H Vincenzo Librandi, <a href="/A003518/b003518.txt">Table of n, a(n) for n = 3..500</a>

%H Daniel Birmajer, Juan B. Gil, Michael D. Weiner, <a href="https://arxiv.org/abs/1707.09918">Bounce statistics for rational lattice paths</a>, arXiv:1707.09918 [math.CO], 2017, p. 9.

%H S. J. Cyvin, J. Brunvoll, E. Brendsdal, B. N. Cyvin and E. K. Lloyd, <a href="http://dx.doi.org/10.1021/ci00026a012">Enumeration of polyene hydrocarbons: a complete mathematical solution</a>, J. Chem. Inf. Comput. Sci., 35 (1995) 743-751.

%H S. J. Cyvin, J. Brunvoll, E. Brendsdal, B. N. Cyvin and E. K. Lloyd, <a href="/A002057/a002057.pdf">Enumeration of polyene hydrocarbons: a complete mathematical solution</a>, J. Chem. Inf. Comput. Sci., 35 (1995) 743-751. [Annotated scanned copy]

%H H. H. Gudmundsson, <a href="http://puma.dimai.unifi.it/21_2/9_Gudmundsson.pdf">Dyck paths, standard Young tableaux, and pattern avoiding permutations</a>, PU. M. A. Vol. 21 (2010), No. 2, pp. 265-284 (see 4.5 p. 280).

%H R. K. Guy, <a href="/A005555/a005555.pdf">Letter to N. J. A. Sloane, May 1990</a>

%H R. K. Guy, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL3/GUY/catwalks.html">Catwalks, Sandsteps and Pascal Pyramids</a>, J. Integer Seqs., Vol. 3 (2000), #00.1.6.

%H V. E. Hoggatt, Jr., <a href="/A005676/a005676.pdf">7-page typed letter to N. J. A. Sloane with suggestions for new sequences</a>, circa 1977.

%H V. E. Hoggatt, Jr. and M. Bicknell, <a href="http://www.fq.math.ca/Scanned/14-5/hoggatt1.pdf">Catalan and related sequences arising from inverses of Pascal's triangle matrices</a>, Fib. Quart., 14 (1976), 395-405.

%H O. Mandelshtam, <a href="https://arxiv.org/abs/1502.00948">Multi-Catalan tableaux and the two-species TASEP</a>, arXiv:1502.00948 [math.CO], 2015.

%H O. Mandelshtam, <a href="http://dx.doi.org/10.4171/AIHPD/30">Multi-Catalan tableaux and the two-species TASEP</a>, Ann. Inst. Henri Poincaré Comb. Phys. Interact. vol. 3 (2016), 321-348, DOI 10.4171/AIHPD/30.

%H L. W. Shapiro, <a href="http://dx.doi.org/10.1016/0012-365X(76)90009-1">A Catalan triangle</a>, Discrete Math. 14 (1976), no. 1, 83-90.

%H L. W. Shapiro, <a href="/A003517/a003517.pdf">A Catalan triangle</a>, Discrete Math. 14 (1976), no. 1, 83-90. [Annotated scanned copy]

%H _Zoran Sunic_, <a href="http://www.combinatorics.org/ojs/index.php/eljc/article/view/v10i1n5">Self describing sequences and the Catalan family tree</a>, Elect. J. Combin., 10 (No. 1, 2003).

%H W.-J. Woan, L. Shapiro and D. G. Rogers, <a href="http://www.jstor.org/stable/2974473">The Catalan numbers, the Lebesgue integral and 4^{n-2}</a>, Amer. Math. Monthly, 104 (1997), 926-931.

%F G.f.: x^3*C(x)^8, where C(x)=(1-sqrt(1-4*x))/(2*x) is g.f. for the Catalan numbers (A000108). - _Emeric Deutsch_, May 30 2004

%F The convolution of A002057 with itself. - _Gerald McGarvey_, Nov 08 2007

%F 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>=7, a(n-4)=(-1)^(n-7)*coeff(charpoly(A,x),x^7). - _Milan Janjic_, Jul 08 2010

%F a(n) = A214292(2*n,n-4) for n > 3. - _Reinhard Zumkeller_, Jul 12 2012

%F Integral representation as the n-th moment of the signed weight function W(x) on (0,4), i.e. in Maple notation: a(n+3) = int(x^n*W(x),x=0..4), n=0,1..., with W(x) = (1/2)*x^(7/2)*(x-2)*(x^2-4*x+2)*sqrt(4-x)/Pi. - _Karol A. Penson_ , Oct 26 2016

%F From _Ilya Gutkovskiy_, Jan 22 2017: (Start)

%F E.g.f.: 4*BesselI(4,2*x)*exp(2*x)/x.

%F a(n) ~ 4^(n+2)/(sqrt(Pi)*n^(3/2)). (End)

%e G.f. = x^3 + 8*x^4 + 44*x^5 + 208*x^6 + 910*x^7 + 3808*x^8 + 15504*x^9 + ...

%t Table[8 Binomial[2 n + 1, n - 3]/(n + 5), {n, 3, 25}] (* _Michael De Vlieger_, Oct 26 2016 *)

%t CoefficientList[Series[((1 - Sqrt[1 - 4 x])/(2 x))^8, {x, 0, 30}], x] (* _Vincenzo Librandi_, Jan 23 2017 *)

%o (PARI) {a(n) = if( n<3, 0, 8 * binomial(2*n + 1, n-3) / (n + 5))}; /* _Michael Somos_, Mar 14 2011 */

%o (PARI) x='x+O('x^50); Vec(x^3*((1-(1-4*x)^(1/2))/(2*x))^8) \\ _Altug Alkan_, Nov 01 2015

%o (MAGMA) [8*Binomial(2*n+1,n-3)/(n+5): n in [3..30]]; // _Vincenzo Librandi_, Jan 23 2017

%Y Cf. A002057.

%Y First differences are in A026018.

%Y A diagonal of any of the essentially equivalent arrays A009766, A030237, A033184, A059365, A099039, A106566, A130020, A047072.

%Y Cf. A000108, A000245, A000344, A000588, A001392, A002057, A003517, A003519.

%K nonn,easy

%O 3,2

%A _N. J. A. Sloane_

%E More terms from _Jon E. Schoenfield_, May 06 2010

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Last modified February 21 19:50 EST 2018. Contains 299423 sequences. (Running on oeis4.)