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A002294 Binomial(5n,n)/(4n+1).
(Formerly M3977 N1646)
50
1, 1, 5, 35, 285, 2530, 23751, 231880, 2330445, 23950355, 250543370, 2658968130, 28558343775, 309831575760, 3390416787880, 37377257159280, 414741863546285, 4628362722856425, 51912988256282175, 584909606696793885, 6617078646960613370 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

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)

REFERENCES

Editor's note: "Über 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.

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).

LINKS

T. D. Noe, Table of n, a(n) for n=0..100

Joerg Arndt, Fxtbook, pp. 337-338.

J. Arndt, Subset-lex: did we miss an order?, arXiv:1405.6503, 2014

Frits Beukers, Hypergeometric functions, how special are they?, Notices Amer. Math. Soc. 61 (2014), no. 1, 48--56. MR3137256

R. W. Gosper, Rope around the earth

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 287

J.-C. Novelli, J.-Y. Thibon, Hopf Algebras of m-permutations,(m+1)-ary trees, and m-parking functions, arXiv preprint arXiv:1403.5962, 2014

Karol A. Penson and Karol Zyczkowski, Product of Ginibre matrices : Fuss-Catalan and Raney distribution, arXiv version

Stillwell, John. Eisenstein's footnote, Math. Intelligencer 17 (1995), no. 2, 58--62. MR1336074 (96d:01024)

Wikipedia, Fuss-Catalan number

S. Yakoubov, Pattern Avoidance in Extensions of Comb-Like Posets, arXiv preprint arXiv:1310.2979, 2013

FORMULA

For the connection with the solution of the quintic, hypergeometric series, and Lagrange inversion, see Beukers (2014). - N. J. A. Sloane, Mar 12 2014

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

EXAMPLE

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 + ...

MATHEMATICA

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 *)

PROG

(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

CROSSREFS

Cf. A002295, A002296, A001764, A002293.

Fourth column of triangle A062993.

Sequence in context: A084135 A229111 A138233 * A051406 A000356 A027392

Adjacent sequences:  A002291 A002292 A002293 * A002295 A002296 A002297

KEYWORD

easy,nonn,nice

AUTHOR

N. J. A. Sloane.

EXTENSIONS

More terms from Olivier Gérard, Jul 05 2001

STATUS

approved

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Last modified July 22 19:50 EDT 2014. Contains 244836 sequences.