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Number of nonseparable toroidal tree-rooted maps with n + 2 edges and n + 1 vertices.
(Formerly M4621)
14

%I M4621 #92 Jun 30 2024 03:25:48

%S 1,9,40,125,315,686,1344,2430,4125,6655,10296,15379,22295,31500,43520,

%T 58956,78489,102885,133000,169785,214291,267674,331200,406250,494325,

%U 597051,716184,853615,1011375,1191640,1396736,1629144,1891505,2186625,2517480,2887221

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

%C The number of faces is 1.

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

%C Sequence of partial sums of A006322. - _L. Edson Jeffery_, Dec 13 2011

%C 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

%D 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.

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

%H Vincenzo Librandi, <a href="/A006414/b006414.txt">Table of n, a(n) for n = 0..10000</a>

%H T. R. S. Walsh and A. B. Lehman, <a href="http://dx.doi.org/10.1016/0095-8956(75)90050-7">Counting rooted maps by genus. III: Nonseparable maps</a>, J. Combinatorial Theory Ser. B, Vol. 18, No. 3 (1975), pp. 222-259.

%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (6,-15,20,-15,6,-1).

%F a(n) = (n+1)*(n+2)^3*(n+3)/24. - _N. J. A. Sloane_, Apr 02 2004

%F a(n) = (n+2)^3*((n+2)^2 - 1)/24. - _Paul Richards_, Mar 04 2007

%F G.f.: (1 + 3*x + x^2)/(1-x)^6. - _Colin Barker_, Feb 21 2012

%F 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

%F a(n) = A143945(n+2)/8. - _J. M. Bergot_, Jun 14 2014

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

%F 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

%F 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

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

%p seq((n+2)^2*binomial(n+3,3)/4, n=0..40); # _G. C. Greubel_, Sep 02 2019

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

%o (Magma) [(n+1)*(n+2)^3*(n+3)/24: n in [0..40]]; // _Wesley Ivan Hurt_, May 10 2014

%o (PARI) a(n) = (n+1)*(n+2)^3*(n+3)/24; \\ _Michel Marcus_, Jul 09 2017

%o (Sage) [(n+2)^2*binomial(n+3,3)/4 for n in (0..40)] # _G. C. Greubel_, Sep 02 2019

%o (GAP) List([0..40], n-> (n+2)^2*Binomial(n+3,3)/4 ); _G. C. Greubel_, Sep 02 2019

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

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

%K nonn,easy

%O 0,2

%A _N. J. A. Sloane_

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

%E Name clarified by _Andrew Howroyd_, Apr 05 2021