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 A002294 Binomial(5n,n)/(4n+1). (Formerly M3977 N1646) 41

%I M3977 N1646

%S 1,1,5,35,285,2530,23751,231880,2330445,23950355,250543370,2658968130,

%T 28558343775,309831575760,3390416787880,37377257159280,

%U 414741863546285,4628362722856425,51912988256282175,584909606696793885

%N Binomial(5n,n)/(4n+1).

%C From _Wolfdieter Lang_, Sep 14 2007: (Start)

%C a(n), n>=1, enumerates quintic trees (rooted, ordered, incomplete) with n vertices (including the root).

%C Pfaff-Fuss-Catalan sequence C^{m}_n for m=5. See the Graham et al. reference, p. 347. eq. 7.66. See also the Pólya-Szegő reference.

%C Also 5-Raney sequence. See the Graham et al. reference, p. 346-7.

%C (End)

%D Editor's note: "Ueber die Bestimmung der Anzahl der verschiedenen Arten, auf welche sich ein n-Eck durch Diagonalen in lauter m-Ecke zerlegen laesst, mit Bezug auf einige Abhandlungen der Herren Lame, Rodrigues, Binet, Catalan und Duhamel in dem Journal de Mathematiques pures et appliquees, publie par Joseph Liouville. T. III. IV.", Archiv der Mathematik u. Physik, 1 (1841), pp. 193ff; see especially p. 198.

%D R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 1990, pp. 200, 347.

%D F. Harary, E. M. Palmer and R. C. Read, On the cell-growth problem for arbitrary polygons, Discr. Math. 11 (1975), 371-389.

%D Karol A. Penson and Karol Zyczkowski, Product of Ginibre matrices: Fuss-Catalan and Raney distributions, Phys. Rev E. vol. 83, 061118 (2011), arXiv:1103.3453, 2011.

%D G. Pólya and G. Szegő, Problems and Theorems in Analysis, Springer-Verlag, Heidelberg, New York, 2 vols., 1972, Vol. 1, problem 211, p. 146 with solution on p. 348.

%D Ulrike Sattler, Decidable classes of formal power series with nice closure properties, Diplomarbeit im Fach Informatik, Univ. Erlangen - Nuernberg, Jul 27 1994

%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).

%D L. Takacs, Enumeration of rooted trees and forests, Math. Scientist 18 (1993), 1-10, esp. Eq. (5).

%H T. D. Noe, <a href="/A002294/b002294.txt">Table of n, a(n) for n=0..100</a>

%H Joerg Arndt, <a href="http://www.jjj.de/fxt/#fxtbook">Fxtbook</a>, pp. 337-338.

%H R. W. Gosper, <a href="http://gosper.org/newsrope.pdf">Rope around the earth</a>

%H INRIA Algorithms Project, <a href="http://algo.inria.fr/ecs/ecs?searchType=1&amp;service=Search&amp;searchTerms=287">Encyclopedia of Combinatorial Structures 287</a>

%H Karol A. Penson and Karol Zyczkowski, <a href="http://dx.doi.org/10.1103/PhysRevE.83.061118">Product of Ginibre matrices : Fuss-Catalan and Raney distribution</a>, <a href="http://arxiv.org/abs/1103.3453/">arXiv version</a>

%F G.f.: hypergeometric([ 1, 2, 3, 4] / 5, [ 2, 3, 5] / 4, x 5^5 / 4^4). - _Michael Somos_, Mar 17 2011

%F O.g.f. A(x) satisfies A(x) = 1 + x * A(x)^5 = 1 / (1 - x * A(x)^4).

%F Given g.f. A(x) then z = t * A(t^4) satisfies 0 = z^5 - z + t. - _Michael Somos_, Mar 17 2011

%F a(n) = binomial(5*n,n-1)/n, n>=1, a(0)=1. From the Lagrange series of the o.g.f. A(x) with its above given implicit equation.

%F a(n) = upper left term in M^n, M = the production matrix:

%F 1, 1

%F 4, 4, 1

%F 10, 10, 4, 1

%F 20, 20, 10, 4, 1

%F ...

%F (where (1, 4, 10, 20,...) is the tetrahedral series, A000292. - _Gary W. Adamson_, Jul 08 2011

%e There are a(2)=5 quintic trees (vertex degree <=5 and 5 possible branchings) with 2 vertices (one of them the root). Adding one more branch (one more vertex) to these five trees yields 5*5+binomial(5,2)=35=a(3) such trees.

%e 1 + x + 5*x^2 + 35*x^3 + 285*x^4 + 2530*x^5 + 23751*x^6 + 231880*x^7 + ...

%e t + t^5 + 5*t^9 + 35*t^13 + 285*t^17 + 2530*t^21 + 23751*t^25 + 231880*t^29 + ...

%t CoefficientList[InverseSeries[ Series[ y - y^5, {y, 0, 100}], x], x][[Range[2, 100, 4]]]

%t Table[Binomial[5n,n]/(4n+1),{n,0,20}] (* _Harvey P. Dale_, Dec 30 2011 *)

%o (PARI) {a(n) = if( n<0, 0, binomial( 5 * n, n) / (n + 1))} /* _Michael Somos_, Mar 17 2011 */

%o (PARI) {a(n) = if( n<0, 0, n = 4*n + 1; polcoeff( serreverse( x - x^5 + x * O(x^n) ), n))} /* _Michael Somos_, Mar 17 2011 */

%o (MAGMA) [ Binomial(5*n,n)/(4*n+1): n in [0..100]]. - _Vincenzo Librandi_, Mar 24 2011

%Y Cf. A002295, A002296, A001764, A002293.

%Y Fourth column of triangle A062993.

%K easy,nonn,nice

%O 0,3

%A _N. J. A. Sloane_.

%E More terms from _Olivier Gérard_, Jul 05 2001

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