A133754 -id:A133754 - cp's OEIS frontend

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

N. J. A. Sloane

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

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).

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).

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

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

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

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

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

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

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

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

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

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

Name clarified by Andrew Howroyd, Apr 05 2021

A284823 Array read by antidiagonals: T(n,k) = number of primitive (aperiodic) palindromes of length n using a maximum of k different symbols (n >= 1, k >= 1).

Original entry on oeis.org

1, 2, 0, 3, 0, 0, 4, 0, 2, 0, 5, 0, 6, 2, 0, 6, 0, 12, 6, 6, 0, 7, 0, 20, 12, 24, 4, 0, 8, 0, 30, 20, 60, 18, 14, 0, 9, 0, 42, 30, 120, 48, 78, 12, 0, 10, 0, 56, 42, 210, 100, 252, 72, 28, 0, 11, 0, 72, 56, 336, 180, 620, 240, 234, 24, 0, 12, 0, 90, 72, 504, 294, 1290, 600, 1008, 216, 62

Offset: 1

Andrew Howroyd, Apr 03 2017

			Table starts:
1  2   3    4    5    6     7     8     9    10 ...
0  0   0    0    0    0     0     0     0     0 ...
0  2   6   12   20   30    42    56    72    90 ...
0  2   6   12   20   30    42    56    72    90 ...
0  6  24   60  120  210   336   504   720   990 ...
0  4  18   48  100  180   294   448   648   900 ...
0 14  78  252  620 1290  2394  4088  6552  9990 ...
0 12  72  240  600 1260  2352  4032  6480  9900 ...
0 28 234 1008 3100 7740 16758 32704 58968 99900 ...
0 24 216  960 3000 7560 16464 32256 58320 99000 ...
...
Row 4 includes palindromes of the form abba but excludes those of the form aaaa, so T(4,k) is k*(k-1).
Row 6 includes palindromes of the forms aabbaa, abbbba, abccba but excludes those of the forms aaaaaa, abaaba, so T(6,k) is 2*k*(k-1) + k*(k-1)*(k-2).

M. R. Nester (1999). Mathematical investigations of some plant interaction designs. PhD Thesis. University of Queensland, Brisbane, Australia. [See A056391 for pdf file of Chap. 2]

Andrew Howroyd, Table of n, a(n) for n = 1..1275

Columns 2-6 are A056458, A056459, A056460, A056461, A056462.
Rows 5-10 are A007531(k+1), A045991, A058895, A047928(k-1), A135497, A133754.
Cf. A284826, A284841.

T[n_, k_] := DivisorSum[n, MoebiusMu[n/#]*k^Ceiling[#/2]&]; Table[T[n-k+1, k], {n, 1, 12}, {k, n, 1, -1}] // Flatten (* Jean-François Alcover, Jun 05 2017 *)

a(n,k) = sumdiv(n, d, moebius(n/d) * k^(ceil(d/2)));
for(n=1, 10, for(k=1, 10, print1( a(n,k),", ");); print();)

T(n,k) = Sum_{d | n} mu(n/d) * k^(ceiling(d/2)).

A253285 a(n) = RF(n+1,3)*C(n+2,n-1), where RF(a,n) is the rising factorial.

Original entry on oeis.org

0, 24, 240, 1200, 4200, 11760, 28224, 60480, 118800, 217800, 377520, 624624, 993720, 1528800, 2284800, 3329280, 4744224, 6627960, 9097200, 12289200, 16364040, 21507024, 27931200, 35880000, 45630000, 57493800, 71823024, 89011440, 109498200, 133771200, 162370560

Offset: 0

Peter Luschny, Mar 23 2015

Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).

Cf. A002378, A006542, A083374, A133754, A253284.

List([0..40], n -> n*((n+1)*(n+2))^2*(n+3)/6); # Bruno Berselli, Mar 06 2018

[n*((n+1)*(n+2))^2*(n+3)/6: n in [0..40]]; // Bruno Berselli, Mar 06 2018

Maple
```
seq(n*((n+1)*(n+2))^2*(n+3)/6,n=0..19);
```

Table[n ((n + 1) (n + 2))^2 (n + 3)/6, {n, 0, 40}] (* Bruno Berselli, Mar 06 2018 *)
LinearRecurrence[{7,-21,35,-35,21,-7,1},{0,24,240,1200,4200,11760,28224},40] (* Harvey P. Dale, Aug 05 2024 *)

[n*((n+1)*(n+2))**2*(n+3)/6 for n in range(40)] # Bruno Berselli, Mar 06 2018

[n*((n+1)*(n+2))^2*(n+3)/6 for n in (0..40)] # Bruno Berselli, Mar 06 2018

G.f.: -24/(x-1)^4 - 144/(x-1)^5 - 240/(x-1)^6 - 120/(x-1)^7. See the comment in A253284 for the general case.
a(n) = n*((n+1)*(n+2))^2*(n+3)/6.
a(n) = (N^3 + 4*N^2 + 4*N)/6 = N*(N + 2)^2/6 with N = n^2 + 3*n.
From Bruno Berselli, Mar 06 2018: (Start)
a(n) = 24*A006542(n+3) for n>0.
a(n) = Sum_{i=0..n} i*(i+1)^3*(i+2). Therefore, the first differences are in A133754. (End)

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

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A284823 Array read by antidiagonals: T(n,k) = number of primitive (aperiodic) palindromes of length n using a maximum of k different symbols (n >= 1, k >= 1).

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A253285 a(n) = RF(n+1,3)*C(n+2,n-1), where RF(a,n) is the rising factorial.

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GAP

Magma

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