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

%I A003518 M4529 #136 May 04 2025 01:54:09
%S A003518 1,8,44,208,910,3808,15504,62016,245157,961400,3749460,14567280,
%T A003518 56448210,218349120,843621600,3257112960,12570420330,48507033744,
%U A003518 187187399448,722477682080,2789279908316,10772391370048,41620603020640,160878516023680,622147386185325
%N A003518 a(n) = 8*binomial(2*n+1,n-3)/(n+5).
%C A003518 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 A003518 Number of standard tableaux of shape (n+4,n-3). - _Emeric Deutsch_, May 30 2004
%D A003518 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A003518 Vincenzo Librandi, <a href="/A003518/b003518.txt">Table of n, a(n) for n = 3..500</a>
%H A003518 Daniel Birmajer, Juan B. Gil and 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 A003518 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., Vol. 35, No. 4 (1995), pp. 743-751.
%H A003518 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., Vol. 35, No. 4 (1995), pp. 743-751. [Annotated scanned copy]
%H A003518 Hilmar Haukur Gudmundsson, <a href="https://arxiv.org/abs/0912.4747">Dyck paths, standard Young tableaux, and pattern avoiding permutations</a>, PU. M. A., Vol. 21, No.2 (2010), pp. 265-284; arXiv:0912.4747 [math.CO], 2009 (see Theorem 11 in Section 4.5).
%H A003518 Richard K. Guy, <a href="/A005555/a005555.pdf">Letter to N. J. A. Sloane, May 1990</a>.
%H A003518 Richard K. Guy, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL3/GUY/catwalks.html">Catwalks, Sandsteps and Pascal Pyramids</a>, J. Integer Seq., Vol. 3 (2000), Article 00.1.6.
%H A003518 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 A003518 V. E. Hoggatt, Jr. and M. Bicknell, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Scanned/14-5/hoggatt1.pdf">Catalan and related sequences arising from inverses of Pascal's triangle matrices</a>, Fib. Quart., Vol. 14, No. 5 (1976), pp. 395-405.
%H A003518 Olya 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 A003518 Olya 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), pp. 321-348, DOI 10.4171/AIHPD/30.
%H A003518 L. W. Shapiro, <a href="http://dx.doi.org/10.1016/0012-365X(76)90009-1">A Catalan triangle</a>, Discrete Math., Vol. 14, No. 1 (1976), pp. 83-90.
%H A003518 L. W. Shapiro, <a href="/A003517/a003517.pdf">A Catalan triangle</a>, Discrete Math., Vol. 14, No. 1 (1976), pp. 83-90. [Annotated scanned copy]
%H A003518 Zoran Sunic, <a href="https://doi.org/10.37236/1745">Self describing sequences and the Catalan family tree</a>, Elect. J. Combin., Vol. 10 (2003), Article N5.
%H A003518 Wen-Jin Woan, Lou 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, Vol. 104, No. 10 (1997), pp. 926-931.
%F A003518 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 A003518 The convolution of A002057 with itself. - _Gerald McGarvey_, Nov 08 2007
%F A003518 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 A003518 a(n) = A214292(2*n,n-4) for n > 3. - _Reinhard Zumkeller_, Jul 12 2012
%F A003518 Integral representation as the n-th moment of the signed weight function W(x) on (0,4), i.e.: a(n+3) = Integral_{x=0..4} x^n*W(x) dx, n >= 0, 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 A003518 From _Ilya Gutkovskiy_, Jan 22 2017: (Start)
%F A003518 E.g.f.: 4*BesselI(4,2*x)*exp(2*x)/x.
%F A003518 a(n) ~ 4^(n+2)/(sqrt(Pi)*n^(3/2)). (End)
%F A003518 D-finite with recurrence: -(n+5)*(n-3)*a(n) +2*n*(2*n+1)*a(n-1)=0. - _R. J. Mathar_, Feb 20 2020
%F A003518 From _Amiram Eldar_, Jan 02 2022: (Start)
%F A003518 Sum_{n>=3} 1/a(n) = 43*Pi/(36*sqrt(3)) - 81/80.
%F A003518 Sum_{n>=3} (-1)^(n+1)/a(n) = 6213*log(phi)/(50*sqrt(5)) - 10339/400, where phi is the golden ratio (A001622). (End)
%e A003518 G.f. = x^3 + 8*x^4 + 44*x^5 + 208*x^6 + 910*x^7 + 3808*x^8 + 15504*x^9 + ...
%t A003518 Table[8 Binomial[2 n + 1, n - 3]/(n + 5), {n, 3, 25}] (* _Michael De Vlieger_, Oct 26 2016 *)
%t A003518 CoefficientList[Series[((1 - Sqrt[1 - 4 x])/(2 x))^8, {x, 0, 30}], x] (* _Vincenzo Librandi_, Jan 23 2017 *)
%o A003518 (PARI) {a(n) = if( n<3, 0, 8 * binomial(2*n + 1, n-3) / (n + 5))}; /* _Michael Somos_, Mar 14 2011 */
%o A003518 (PARI) my(x='x+O('x^50)); Vec(x^3*((1-(1-4*x)^(1/2))/(2*x))^8) \\ _Altug Alkan_, Nov 01 2015
%o A003518 (Magma) [8*Binomial(2*n+1,n-3)/(n+5): n in [3..30]]; // _Vincenzo Librandi_, Jan 23 2017
%Y A003518 Cf. A002057.
%Y A003518 First differences are in A026018.
%Y A003518 A diagonal of any of the essentially equivalent arrays A009766, A030237, A033184, A059365, A099039, A106566, A130020, A047072.
%Y A003518 Cf. A000108, A000245, A000344, A000588, A001392, A002057, A003517, A003519, A001622.
%K A003518 nonn,easy
%O A003518 3,2
%A A003518 _N. J. A. Sloane_
%E A003518 More terms from _Jon E. Schoenfield_, May 06 2010