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A006298 Number of genus 2 rooted maps with 1 face with n vertices.
(Formerly M5117)
13
21, 483, 6468, 66066, 570570, 4390386, 31039008, 205633428, 1293938646, 7808250450, 45510945480, 257611421340, 1422156202740, 7683009544980, 40729207226400, 212347275857640, 1090848505817070, 5530195966465170, 27704671055301240, 137308238124957900, 673903972248687180, 3278143051447003740, 15816495077491530240, 75740811006275677080, 360195962116311020700, 1702004224469594857812, 7994567449203067400976, 37343992994700814841496, 173539732151844963086952, 802554981295852197252840, 3694707104076119563303872, 16936911943685345325329616 (list; graph; refs; listen; history; text; internal format)
OFFSET

4,1

REFERENCES

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. R. S. Walsh, Combinatorial Enumeration of Non-Planar Maps. Ph.D. Dissertation, Univ. of Toronto, 1971.

Liang Zhao and Fengyao Yan, Note on Total Positivity for a Class of Recursive Matrices, Journal of Integer Sequences, Vol. 19 (2016), Article 16.6.5.

LINKS

G. C. Greubel, Table of n, a(n) for n = 4..1000

Robert Cori, G Hetyei, Counting partitions of a fixed genus, arXiv preprint arXiv:1710.09992, 2017

T. R. S. Walsh and A. B. Lehman, Counting rooted maps by genus, J. Comb. Thy B13 (1972), 122-141 and 192-218.

FORMULA

a(n+1) = ((5*n+3)*(4n*+2)*a(n))/((5*n-2)(n-3)).

G.f.: 21*x^4*(1+x)/sqrt((1-4*x)^11). a(n) = 21 * (A020922(n-4) + A020922(n-3)). - Ralf Stephan, Mar 13 2004 (g.f. corrected by Joerg Arndt, Apr 07 2013)

0 = a(n)*(+16*a(n+1) +62*a(n+2) +6*a(n+3)) +a(n+1)*(-38*a(n+1) -5*a(n+2) +17*a(n+3)) +a(n+2)*(-23*a(n+2) +a(n+3)) for all n in Z. - Michael Somos, Mar 30 2016

a(n) ~ n^(9/2) * 2^(2*n-5) / (9*sqrt(Pi)). - Vaclav Kotesovec, Mar 30 2016

EXAMPLE

G.f. = 21*x^4 + 483*x^5 + 6468*x^6 + 66066*x^7 + 570570*x^8 + 4390386*x^9 + ...

MATHEMATICA

CoefficientList[Series[21*x^4*(1 + x)/Sqrt[(1 - 4*x)^11], {x, 0, 50}]/x^4, x] (* G. C. Greubel, Jan 30 2017 *)

PROG

(PARI) A006298(n) = if(n<4, 0, if(n==4, 21, ((5*(n-1)+3)*(4*(n-1)+2)*A006298(n-1))/((5*(n-1)-2)*((n-1)-3)))); \\ Joerg Arndt, Apr 07 2013

(PARI) x='x+O('x^66);  Vec(21*x^4*(1+x)/sqrt((1-4*x)^11)) \\ Joerg Arndt, Apr 07 2013

CROSSREFS

Cf. A035309.

Sequence in context: A025603 A296586 A269922 * A089907 A015695 A006299

Adjacent sequences:  A006295 A006296 A006297 * A006299 A006300 A006301

KEYWORD

nonn,easy

AUTHOR

N. J. A. Sloane

STATUS

approved

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Last modified April 26 03:33 EDT 2018. Contains 303098 sequences. (Running on oeis4.)