A006414 Number of nonseparable toroidal tree-rooted maps with n + 2 edges and n + 1 vertices.

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

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

The number of faces is 1.

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

The sequence b(n) = a(n-2) with a(-1) = 0, for n >= 1, is b(n) = n^3*(n^2 - 1)/4!. It is obtained by comparing the result for the powers n^5 from Worpitzky's identity (see a formula in A000584) with the result obtained from the counting of degrees of freedom for the decomposition of a rank 5 tensor in n dimensions via the standard Young tableaux version with 5 boxes corresponding to the seven partitions of 5. The difference of the two versions gives: 10*(binomial(n+3, 5) + 3*binomial(n+2, 5) + binomial(n+1, 5)) = 5*n*(binomial(n+2, 4) + binomial(n+1, 4)) = 10*b(n). See the formula for a(n) below. - Wolfdieter Lang, Jul 18 2019

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), and p. 186.
• 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 and A. B. Lehman, Counting rooted maps by genus. III: Nonseparable maps, J. Combinatorial Theory Ser. B, Vol. 18, No. 3 (1975), pp. 222-259.
• Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

Crossrefs

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

Cf. A004068, A005891, A006322, A006415, A006434, A006441, A133754, A143945.

Programs

GAP

List([0..40], n-> (n+2)^2*Binomial(n+3,3)/4 ); G. C. Greubel, Sep 02 2019

Magma

[(n+1)*(n+2)^3*(n+3)/24: n in [0..40]]; // Wesley Ivan Hurt, May 10 2014

Maple

seq((n+2)^2*binomial(n+3,3)/4, n=0..40); # G. C. Greubel, Sep 02 2019

Mathematica

Table[(n + 1)*(n + 2)^3*(n + 3)/24, {n, 0, 40}] (* Vladimir Joseph Stephan Orlovsky, Apr 18 2011 *)

PARI

a(n) = (n+1)*(n+2)^3*(n+3)/24; \\ Michel Marcus, Jul 09 2017

Sage

[(n+2)^2*binomial(n+3,3)/4 for n in (0..40)] # G. C. Greubel, Sep 02 2019

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) = (Sum_{k=0..n+1} k*(n+1)*((n+1)^2 - k^2))/6 for n > 0, 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

Sum_{n>=0} 1/a(n) = 30 - 24*zeta(3). - Jaume Oliver Lafont, Jul 09 2017

a(n) = binomial(n+5, 5) + 3*binomial(n+4, 5) + binomial(n+3, 5) = ((n+2)/2)*(binomial(n+4, 4) + binomial(n+3, 4)), for n >= 0. See a comment above on the sequence b(n) = a(n-2) = n^3*(n^2 - 1)/4!. - Wolfdieter Lang, Jul 19 2019

E.g.f.: (24 + 192*x + 276*x^2 + 124*x^3 + 20*x^4 + x^5)*exp(x)/4!. - G. C. Greubel, Sep 02 2019

Sum_{n>=0} (-1)^n/a(n) = 18*zeta(3) + 48*log(2) - 54. - Amiram Eldar, Jan 09 2022

Extensions

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

Name clarified by Andrew Howroyd, Apr 05 2021