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A001392 a(n) = 9*binomial(2n,n-4)/(n+5).
(Formerly M4637 N1981)
25

%I M4637 N1981

%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

%C a(n) = A214292(2*n-1,n-5) for n > 4. - _Reinhard Zumkeller_, Jul 12 2012

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

%H A. Papoulis, <a href="/A000108/a000108_8.pdf">A new method of inversion of the Laplace transform</a>, Quart. Appl. Math 14 (1957), 405-414. [Annotated scan of selected pages]

%H A. Papoulis, <a href="http://www.jstor.org/stable/43636019">A new method of inversion of the Laplace transform</a>, Quart. Applied Math. 14 (1956), 405ff.

%H J. 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., 29 (1975), 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, 10 (2003) #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 -(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)

%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.

%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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Last modified October 17 03:15 EDT 2018. Contains 316275 sequences. (Running on oeis4.)