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A002294
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Binomial(5n,n)/(4n+1).
(Formerly M3977 N1646)
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41
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1, 1, 5, 35, 285, 2530, 23751, 231880, 2330445, 23950355, 250543370, 2658968130, 28558343775, 309831575760, 3390416787880, 37377257159280, 414741863546285, 4628362722856425, 51912988256282175, 584909606696793885
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OFFSET
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0,3
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COMMENTS
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From Wolfdieter Lang, Sep 14 2007: (Start)
a(n), n>=1, enumerates quintic trees (rooted, ordered, incomplete) with n vertices (including the root).
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.
Also 5-Raney sequence. See the Graham et al. reference, p. 346-7.
(End)
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REFERENCES
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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.
R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 1990, pp. 200, 347.
F. Harary, E. M. Palmer and R. C. Read, On the cell-growth problem for arbitrary polygons, Discr. Math. 11 (1975), 371-389.
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.
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.
Ulrike Sattler, Decidable classes of formal power series with nice closure properties, Diplomarbeit im Fach Informatik, Univ. Erlangen - Nuernberg, Jul 27 1994
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
L. Takacs, Enumeration of rooted trees and forests, Math. Scientist 18 (1993), 1-10, esp. Eq. (5).
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LINKS
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T. D. Noe, Table of n, a(n) for n=0..100
Joerg Arndt, Fxtbook, pp. 337-338.
R. W. Gosper, Rope around the earth
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 287
Karol A. Penson and Karol Zyczkowski, Product of Ginibre matrices : Fuss-Catalan and Raney distribution, arXiv version
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FORMULA
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G.f.: hypergeometric([ 1, 2, 3, 4] / 5, [ 2, 3, 5] / 4, x 5^5 / 4^4). - Michael Somos, Mar 17 2011
O.g.f. A(x) satisfies A(x) = 1 + x * A(x)^5 = 1 / (1 - x * A(x)^4).
Given g.f. A(x) then z = t * A(t^4) satisfies 0 = z^5 - z + t. - Michael Somos, Mar 17 2011
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.
a(n) = upper left term in M^n, M = the production matrix:
1, 1
4, 4, 1
10, 10, 4, 1
20, 20, 10, 4, 1
...
(where (1, 4, 10, 20,...) is the tetrahedral series, A000292. - Gary W. Adamson, Jul 08 2011
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EXAMPLE
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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.
1 + x + 5*x^2 + 35*x^3 + 285*x^4 + 2530*x^5 + 23751*x^6 + 231880*x^7 + ...
t + t^5 + 5*t^9 + 35*t^13 + 285*t^17 + 2530*t^21 + 23751*t^25 + 231880*t^29 + ...
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MATHEMATICA
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CoefficientList[InverseSeries[ Series[ y - y^5, {y, 0, 100}], x], x][[Range[2, 100, 4]]]
Table[Binomial[5n, n]/(4n+1), {n, 0, 20}] (* Harvey P. Dale, Dec 30 2011 *)
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PROG
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(PARI) {a(n) = if( n<0, 0, binomial( 5 * n, n) / (n + 1))} /* Michael Somos, Mar 17 2011 */
(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 */
(MAGMA) [ Binomial(5*n, n)/(4*n+1): n in [0..100]]. - Vincenzo Librandi, Mar 24 2011
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CROSSREFS
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Cf. A002295, A002296, A001764, A002293.
Fourth column of triangle A062993.
Sequence in context: A087630 A084135 A138233 * A051406 A000356 A027392
Adjacent sequences: A002291 A002292 A002293 * A002295 A002296 A002297
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KEYWORD
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easy,nonn,nice
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AUTHOR
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N. J. A. Sloane.
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EXTENSIONS
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More terms from Olivier Gérard, Jul 05 2001
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STATUS
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approved
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