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A006414 Number of nonseparable toroidal tree-rooted maps on n nodes.
(Formerly M4621)
11
1, 9, 40, 125, 315, 686, 1344, 2430, 4125, 6655, 10296, 15379, 22295, 31500, 43520, 58956, 78489, 102885, 133000, 169785, 214291, 267674, 331200, 406250, 494325, 597051, 716184, 853615, 1011375, 1191640, 1396736, 1629144, 1891505, 2186625, 2517480, 2887221 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

a(n) = K(Oa(2,3,n)), Kekulé numbers of certain benzenoid structures (see the Cyvin - Gutman reference).

Sequence of partial sums of A006322. - L. Edson Jeffery, Dec 13 2011

REFERENCES

S. J. Cyvin and I. Gutman, Kekulé structures in benzenoid hydrocarbons, Lecture Notes in Chemistry, No. 46, Springer, New York, 1988, p. 105, eq. (ii). 187).

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

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..10000

T. R. S. Walsh, A. B. Lehman, Counting rooted maps by genus. III: Nonseparable maps, J. Combinatorial Theory Ser. B 18 (1975), 222-259.

FORMULA

a(n) = (n+1)*(n+2)^3*(n+3)/24. - N. J. A. Sloane, Apr 02 2004

a(n) = (n+2)^3((n+2)^2-1)/24. - Paul Richards, Mar 04 2007

G.f.: (1+3*x+x^2)/(1-x)^6. - Colin Barker, Feb 21 2012

a(n) = for n>0 sum[k*(n+1)*((n+1)^2 - k^2)){0<k<n+1}]/6, which is the sum of all areas of Pythagorean triangles with arms 2*k*(n+1) and (n+1)^2 - k^2 with hypotenuse k^2 +(n+1)^2. - J. M. Bergot, May 12 2014

a(n) = A143945(n+2)/8. - J. M. Bergot, Jun 14 2014

MAPLE

a:=n->sum(sum(sum((n-k)*k/4, j=1..n), k=1..n), m=1..n): seq(a(n), n=2..37); # Zerinvary Lajos, May 13 2007

with(combinat):a:=n->sum(sum(sum(binomial(n+2, 2)/12, j=1..n), k=0..n), m=0..n): seq(a(n), n=1..36); # Zerinvary Lajos, May 30 2007

a:=n->sum(n^4-n^3, j=0..n): seq(a(n)/24, n=2..37); # Zerinvary Lajos, May 08 2008

MATHEMATICA

Table[(n + 1)*(n + 2)^3*(n + 3)/24, {n, 0, 80}] (* Vladimir Joseph Stephan Orlovsky, Apr 18 2011 *)

PROG

(MAGMA) [(n+1)*(n+2)^3*(n+3)/24: n in [0..30]]; // Wesley Ivan Hurt, May 10 2014

CROSSREFS

Differences of A006542 (C(n, 3)*C(n-1, 3)/4).

Cf. A005891, A006322, A004068, A143945, A133754.

Sequence in context: A165372 A264624 A287324 * A213758 A181960 A219511

Adjacent sequences:  A006411 A006412 A006413 * A006415 A006416 A006417

KEYWORD

nonn,easy

AUTHOR

N. J. A. Sloane

EXTENSIONS

More terms from Robert Newstedt (Patternfinder(AT)webtv.net)

STATUS

approved

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Last modified June 25 09:56 EDT 2017. Contains 288709 sequences.