Original entry on oeis.org
1, 3, 7, 21, 78, 390, 2461, 17491, 135226, 1103076, 9371892, 82205622, 740254762, 6814312822, 63920746639, 609452784251, 5894288690288, 57728196873452, 571747727911362, 5719672404523644, 57737110684330278, 587604181217075742
Offset: 0
a(21) = 1 + 2 + 4 + 14 + 57 + 312 + 2071 + 15030 + 117735 + 967850 + 8268816 + 72833730 + 658049140 + 6074058060 + 57106433817 + 545532037612 + 5284835906037 + 51833908183164 + 514019531037910 + 5147924676612282 + 52017438279806634 + 529867070532745464.
-
q[n_?OddQ]:= 3^((n-1)/2)*CatalanNumber[(n-1)/2];
q[n_?EvenQ]:= 3^((n-2)/2)*(2*(n-1)/(n+2))*CatalanNumber[(n-2)/2];
f[n_]:= f[n]= Sum[EulerPhi[n/k]*3^k*Binomial[2*k, k], {k, Most[Divisors[n]]}];
A006384[n_]:= If[n==0, 1, (1/(2*n))*(2*(3^n/(n+2))*CatalanNumber[n] +f[n] + 2*n*q[n])];
Table[Sum[A006384[j], {j,0,n}], {n,0,50}] (* G. C. Greubel, Jul 14 2021 *)
A006385
Number of unsensed planar maps with n edges.
Original entry on oeis.org
1, 2, 4, 14, 52, 248, 1416, 9172, 66366, 518868, 4301350, 37230364, 333058463, 3057319072, 28656583950, 273298352168, 2645186193457, 25931472185976, 257086490694917, 2574370590192556, 26010904915620261
Offset: 0
- N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
- T. R. S. Walsh, personal communication.
- Richard Kapolnai, Gabor Domokos, and Timea Szabo, Generating spherical multiquadrangulations by restricted vertex splittings and the reducibility of equilibrium classes, Periodica Polytechnica Electrical Engineering, 56(1):11-10, 2012. Also arXiv:1206.1698, 2012. See Table 2.
- Valery. A. Liskovets, A reductive technique for enumerating nonisomorphic planar maps, Discr. Math., v.156 (1996), 197-217.
- Timothy R. Walsh, Generating nonisomorphic maps without storing them, SIAM J. Algebraic Discrete Methods 4 (1983), no. 2, 161-178.
- Timothy R. Walsh, Space-efficient generation of nonisomorphic maps and hypermaps
- Timothy R. Walsh, Space-Efficient Generation of Nonisomorphic Maps and Hypermaps, J. Int. Seq. 18 (2015) # 15.4.3
- Nicholas C. Wormald, Counting unrooted planar maps, Discrete Math. 36 (1981), no. 2, 205-225.
A379438
Triangle read by rows: T(n,k) is the number of sensed combinatorial maps with n edges and genus k, 0 <= k <= floor(n/2).
Original entry on oeis.org
1, 2, 4, 1, 14, 6, 57, 46, 4, 312, 452, 106, 2071, 4852, 2382, 131, 15030, 52972, 46680, 8158, 117735, 587047, 830848, 313611, 14118, 967850, 6550808, 13804864, 9326858, 1369446, 8268816, 73483256, 218353000, 236095958, 74803564, 2976853, 72833730, 827801468, 3328822880, 5345316004, 3023693380, 391288854
Offset: 0
Triangle begins:
n\k [0] [1] [2] [3] [4]
[0] 1;
[1] 2;
[2] 4, 1;
[3] 14, 6;
[4] 57, 46, 4;
[5] 312, 452, 106;
[6] 2071, 4852, 2382, 131;
[7] 15030, 52972, 46680, 8158;
[8] 117735, 587047, 830848, 313611, 14118;
[9] 967850, 6550808, 13804864, 9326858, 1369446;
...
- Andrew Howroyd, Table of n, a(n) for n = 0..120 (rows 0..20)
- Antonio Breda d'Azevedo, Alexander Mednykh and Roman Nedela, Enumeration of maps regardless of genus: Geometric approach, Discrete Mathematics, Volume 310, 2010, Pages 1184-1203.
- Timothy R. Walsh, Alain Giorgetti, and Alexander Mednykh, Enumeration of unrooted orientable maps of arbitrary genus by number of edges and vertices, Discrete Math. 312 (2012), no. 17, 2660--2671. MR2935417.
Columns 0..10 are
A006384,
A006386,
A104595,
A104596,
A215019,
A239918,
A239919,
A239921,
A239922,
A239923,
A239924.
A006386
Number of sensed genus 1 maps with n edges.
Original entry on oeis.org
1, 6, 46, 452, 4852, 52972, 587047, 6550808, 73483256, 827801468, 9360123740, 106189359544, 1208328304864, 13787042250528, 157700137398689, 1807893066408464, 20768681225892328, 239037464947999900, 2755989928117365244, 31826208029615881656, 368074022535205870382
Offset: 2
- N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
- T. D. Noe, Table of n, a(n) for n = 2..30 (from Mednykh and Nedela)
- A. D. Mednykh and R. Nedela, Enumeration of unrooted maps with given genus, J. Combin. Th. B, 96 (2006), 706-729.
- Timothy R. Walsh, Generating nonisomorphic maps without storing them, SIAM J. Algebraic Discrete Methods 4 (1983), no. 2, 161-178.
- Timothy R. Walsh, Counting maps on doughnuts, Theoretical Computer Science, vol.502, pp.4-15, (September-2013).
- Timothy R. S. Walsh, Alain Giorgetti, Alexander Mednykh, Enumeration of unrooted orientable maps of arbitrary genus by number of edges and vertices, Discrete Math. 312 (2012), no. 17, 2660--2671. MR2935417. - From _N. J. A. Sloane_, Aug 01 2012
A379430
Array read by antidiagonals: A(n,k) is the number of sensed planar maps with n vertices and k faces, n >= 1, k >= 1.
Original entry on oeis.org
1, 1, 1, 1, 2, 1, 2, 5, 5, 2, 3, 14, 23, 14, 3, 6, 42, 108, 108, 42, 6, 14, 140, 501, 761, 501, 140, 14, 34, 473, 2264, 4744, 4744, 2264, 473, 34, 95, 1670, 10087, 27768, 38495, 27768, 10087, 1670, 95, 280, 5969, 44310, 153668, 279698, 279698, 153668, 44310, 5969, 280
Offset: 1
Array begins:
=========================================================
n\k | 1 2 3 4 5 6 7 8 ...
----+----------------------------------------------------
1 | 1 1 1 2 3 6 14 34 ...
2 | 1 2 5 14 42 140 473 1670 ...
3 | 1 5 23 108 501 2264 10087 44310 ...
4 | 2 14 108 761 4744 27768 153668 ...
5 | 3 42 501 4744 38495 279698 ...
6 | 6 140 2264 27768 279698 ...
7 | 14 473 10087 153668 ...
8 | 34 1670 44310 ...
...
As a triangle, rows give the number of edges (first row is 0 edges):
1;
1, 1;
1, 2, 1;
2, 5, 5, 2;
3, 14, 23, 14, 3;
6, 42, 108, 108, 42, 6;
14, 140, 501, 761, 501, 140, 14;
34, 473, 2264, 4744, 4744, 2264, 473, 34;
95, 1670, 10087, 27768, 38495, 27768, 10087, 1670, 95;
...
A006443
Number of achiral planar maps with n edges.
Original entry on oeis.org
1, 2, 4, 14, 47, 184, 761, 3314, 14997, 69886, 333884, 1626998, 8067786, 40580084, 206734083, 1064666724, 5536480877, 29036188788, 153450351924, 816503772830, 4371551433888
Offset: 0
- N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
- V. A. Liskovets, Some easily derivable sequences, J. Integer Sequences, 3 (2000), #00.2.2.
- Timothy R. Walsh, Number of sensed planar maps with n edges and m vertices, p. 43.
- Nicholas C. Wormald, Counting unrooted planar maps, Discrete Math. 36 (1981), no. 2, 205-225.
A006388
Number of sensed planar maps with n edges and without faces of degree 1.
Original entry on oeis.org
1, 1, 2, 6, 18, 74, 393, 2282, 14700, 99614, 703519, 5123598, 38279496
Offset: 0
- N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
A006390
Number of sensed loopless planar maps with n edges.
Original entry on oeis.org
1, 1, 2, 5, 14, 49, 240, 1259, 7570, 47996, 319518, 2199295, 15571610, 112773478, 832809504, 6253673323, 47650870538, 367784975116, 2871331929096, 22647192990256, 180277915464664, 1447060793168493, 11703567787559680, 95312765368320637, 781151020141584190
Offset: 0
- N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
- Andrew Howroyd, Table of n, a(n) for n = 0..500
- V. A. Liskovets and T. R. S. Walsh, Counting Unrooted Loopless Planar Maps [Extended abstract]
- V. A. Liskovets and T. R. S. Walsh, Counting unrooted loopless planar maps, Europ. J. Combin., 26:5 (2005), 651-663.
- Timothy R. Walsh, Generating nonisomorphic maps without storing them, SIAM J. Algebraic Discrete Methods 4 (1983), no. 2, 161-178.
-
a[n_] := If[n==0, 1, (1/(2n))(Sum[Binomial[4k, k] EulerPhi[n/k] Boole[ 0Jean-François Alcover, Aug 29 2019 *)
-
a(n) = {if(n==0, 1, (sumdiv(n, d, if(dAndrew Howroyd, Mar 28 2021
A006394
Number of sensed planar maps with n edges and without loops or parallel edges.
Original entry on oeis.org
1, 1, 1, 3, 5, 15, 52, 213, 1002, 5167, 27967, 158447, 926786
Offset: 0
- N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
A006392
Number of sensed planar maps with n edges and without faces of degree 1 or 2.
Original entry on oeis.org
1, 0, 1, 4, 9, 34, 161, 830, 4779, 29092, 184510, 1208178, 8116922
Offset: 0
- N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
Showing 1-10 of 17 results.
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