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%I M4637 N1981 #72 Jan 02 2022 09:48:48
%S 1,9,54,273,1260,5508,23256,95931,389367,1562275,6216210,24582285,
%T 96768360,379629720,1485507600,5801732460,22626756594,88152205554,
%U 343176898988,1335293573130,5193831553416,20198233818840,78542105700240,305417807763705
%N a(n) = 9*binomial(2n,n-4)/(n+5).
%C Number of n-th generation vertices in the tree of sequences with unit increase labeled by 8 (cf. _Zoran Sunic_ reference) - _Benoit Cloitre_, Oct 07 2003
%C 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=4. - _Herbert Kociemba_, May 24 2004
%C Number of standard tableaux of shape (n+4,n-4). - _Emeric Deutsch_, May 30 2004
%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H T. D. Noe, <a href="/A001392/b001392.txt">Table of n, a(n) for n = 4..200</a>
%H Richard K. Guy, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL3/GUY/catwalks.html">Catwalks, sandsteps and Pascal pyramids</a>, J. Integer Sequences, Vol. 3 (2000), Article #00.1.6.
%H Athanasios Papoulis, <a href="/A000108/a000108_8.pdf">A new method of inversion of the Laplace transform</a>, Quart. Appl. Math., Vol. 14 (1957), pp. 405-414. [Annotated scan of selected pages]
%H Athanasios Papoulis, <a href="http://www.jstor.org/stable/43636019">A new method of inversion of the Laplace transform</a>, Quart. Applied Math., Vol. 14 (1957), pp. 405-414.
%H John Riordan, <a href="https://doi.org/10.1090/S0025-5718-1975-0366686-9">The distribution of crossings of chords joining pairs of 2n points on a circle</a>, Math. Comp., Vol. 29, No. 129 (1975), pp. 215-222.
%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>, Electronic Journal of Combinatorics, Vol. 10 (2003), Article #N5.
%F Expansion of x^4*C^9, where C = (1-(1-4*x)^(1/2))/(2*x) is g.f. for Catalan numbers, A000108. - _Philippe Deléham_, Feb 03 2004
%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>=8, a(n-4)=(-1)^(n-8)*coeff(charpoly(A,x),x^8). - _Milan Janjic_, Jul 08 2010
%F a(n) = A214292(2*n-1,n-5) for n > 4. - _Reinhard Zumkeller_, Jul 12 2012
%F D-finite with recurrence -(n+5)*(n-4)*a(n) +2*n*(2*n-1)*a(n-1)=0. - _R. J. Mathar_, Jun 20 2013
%F From _Ilya Gutkovskiy_, Jan 22 2017: (Start)
%F E.g.f.: (1/24)*x^4*1F1(9/2; 10; 4*x).
%F a(n) ~ 9*4^n/(sqrt(Pi)*n^(3/2)). (End)
%F From _Amiram Eldar_, Jan 02 2022: (Start)
%F Sum_{n>=4} 1/a(n) = 158*Pi/(81*sqrt(3)) - 649/270.
%F Sum_{n>=4} (-1)^n/a(n) = 52076*log(phi)/(225*sqrt(5)) - 22007/450, where phi is the golden ratio (A001622). (End)
%e G.f. = x^4 + 9*x^5 + 54*x^6 + 273*x^7 + 1260*x^8 + 5508*x^9 + 23256*x^10 + ...
%p A001392:=n->9*binomial(2*n,n-4)/(n+5): seq(A001392(n), n=4..40); # _Wesley Ivan Hurt_, Apr 11 2017
%t Table[9*Binomial[2n,n-4]/(n+5),{n,4,30}] (* _Harvey P. Dale_, Mar 03 2011 *)
%o (PARI) a(n)=9*binomial(n+n,n-4)/(n+5) \\ _Charles R Greathouse IV_, Jul 31 2011
%Y First differences are in A026015.
%Y A diagonal of any of the essentially equivalent arrays A009766, A030237, A033184, A059365, A099039, A106566, A130020, A047072.
%Y Cf. A000108, A000245, A002057, A000344, A003517, A000588, A003518, A003519, A001622.
%K nonn,easy
%O 4,2
%A _N. J. A. Sloane_
%E More terms from _Harvey P. Dale_, Mar 03 2011
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