A007401 -id:A007401 - cp's OEIS frontend

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.

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

Andrew Howroyd, Jan 13 2025

The planar maps considered are connected and may contain loops and parallel edges.

The number of edges is n + k - 2.

			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;
  ...

Timothy R. Walsh, Number of sensed planar maps with n edges and m vertices, pp. 1-10.

Antidiagonal sums are A006384.
Columns 1..2 are A002995, A380237.
Cf. A269920 (rooted), A277741 (unsensed), A379431 (achiral), A342061 (2-connected), A384964 (simple).

A(n,k) = A(k,n).

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

An achiral map is a map with a sense-reversing automorphism.

The planar maps considered are connected and may contain loops and parallel edges. - Andrew Howroyd, Jan 13 2025

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.

Column k=0 of A380234.

Cf. A006384 (sensed), A006385 (unsensed), A006444 (2-connected), A006445 (3-connected).

a(n) = 2*A006385(n) - A006384(n). [Liskovets eq 3a] - R. J. Mathar, Oct 01 2011

a(0)=1 prepended by Andrew Howroyd, Jan 13 2025

a(20) added by Andrew Howroyd, Jan 20 2025

A006403 Number of unsensed 2-connected planar maps with n edges.

Original entry on oeis.org

0, 1, 2, 3, 6, 15, 36, 114, 396, 1565, 6756, 31563, 154370, 785113, 4099948, 21870704, 118624544, 652485364, 3631820462, 20426666644, 115949791342, 663640383400, 3826858500878, 22218232389849, 129802836253994

Offset: 1

N. J. A. Sloane

nonn

The maps considered here may include parallel edges. - Andrew Howroyd, Jan 13 2025

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

Timothy R. Walsh, Generating nonisomorphic maps without storing them, SIAM J. Algebraic Discrete Methods 4 (1983), no. 2, 161-178.
Timothy R. Walsh, Number of sensed planar maps with n edges and m vertices, p. 41.

Row sums of A379432.

Cf. A000139 (rooted), A006385, A006402 (sensed), A006407 (without parallel edges), A006444 (achiral).

a(n) = (A006402(n) + A006444(n))/2. - Andrew Howroyd, Jan 13 2025

a(23)-a(25) added by Andrew Howroyd, Jan 13 2025

A001853 Total height of trees with n nodes.

Original entry on oeis.org

0, 1, 3, 8, 22, 58, 158, 425, 1161, 3175, 8751, 24192, 67239, 187459, 524579, 1472086, 4142639, 11685501, 33036492, 93583873, 265589158, 754998424, 2149588878, 6128914829, 17497927510, 50017546412, 143137957590, 410062891934, 1175925220607, 3375322972597

Offset: 1

N. J. A. Sloane

nonn

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Alois P. Heinz, Table of n, a(n) for n = 1..300
J. Riordan, Enumeration of trees by height and diameter, IBM J. Res. Dev. 4 (1960), 473-478.
J. Riordan, The enumeration of trees by height and diameter, IBM Journal 4 (1960), 473-478. (Annotated scanned copy)
Index entries for sequences related to trees

Cf. A034781.

a(n) = Sum_{k=1..n-1} A034781(n,k)*k. - Sean A. Irvine, Mar 24 2016

More terms from Sean A. Irvine, Mar 24 2016

A000184 Number of genus 0 rooted maps with 3 faces with n vertices.

Original entry on oeis.org

2, 22, 164, 1030, 5868, 31388, 160648, 795846, 3845020, 18211380, 84876152, 390331292, 1775032504, 7995075960, 35715205136, 158401506118, 698102372988, 3059470021316, 13341467466520, 57918065919924, 250419305769512, 1078769490401032, 4631680461623664, 19825379450255900, 84622558822506328, 360270317908904328, 1530148541536781488, 6484511936352543096, 27423786092731382000, 115756362341775227888

Offset: 2

N. J. A. Sloane

nonn

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
T. R. S. Walsh, Combinatorial Enumeration of Non-Planar Maps. Ph.D. Dissertation, Univ. of Toronto, 1971.

Alois P. Heinz, Table of n, a(n) for n = 2..500
Richard P. Stanley, CATALAN ADDENDUM, version of Jul 19, 2008, p. 24. [From _Jonathan Vos Post_, Aug 16 2008]
M. S. Tokmachev, Correlations Between Elements and Sequences in a Numerical Prism, Bulletin of the South Ural State University, Ser. Mathematics. Mechanics. Physics, 2019, Vol. 11, No. 1, 24-33.
W. T. Tutte, On the enumeration of planar maps, Bull. Amer. Math. Soc. 74 1968 64-74.
T. R. S. Walsh and A. B. Lehman, Counting rooted maps by genus, J. Comb. Thy B13 (1972), 122-141 and 192-218.
Notes

Column 3 of A269920.
Column 0 of A270407.
Cf. A000108, A029887.

[n*((n+1)*(n+2)*Catalan(n+1) - 3*4^n)/12: n in [2..30]]; // G. C. Greubel, Jul 18 2024

a[n_] := 1/12*(2^(n+1)*(2*n+1)!!/(n-1)!-3*4^n*n); Table[a[n], {n, 2, 31}] (* Jean-François Alcover, Mar 12 2014 *)

[n*(2*(2*n+1)*binomial(2*n,n) - 3*4^n)//12 for n in range(2,30)] # G. C. Greubel, Jul 18 2024

a(n) = 2 * A029887(n-2). - Ralf Stephan, Aug 17 2004
a(n) = 4^n*Gamma(n+3/2)/(3*sqrt(Pi)*Gamma(n)) - n*4^(n-1). - Mark van Hoeij, Jul 06 2010
From G. C. Greubel, Jul 18 2024: (Start)
a(n) = (n/12)*( (n+1)*(n+2)*Catalan(n+1) - 3*4^n ).
G.f.: x*(1 - sqrt(1 - 4*x))/(1-4*x)^(5/2).
E.g.f.: (x/3)*exp(2*x)*( - 3*exp(2*x) + 3*(1+2*x)*BesselI(0, 2*x) + (3+8*x)*BesselI(1, 2*x) + 2*x*BesselI(2, 2*x) ). (End)

More terms from Sean A. Irvine, Nov 14 2010

A004139 Odd primes excluding 5.

Original entry on oeis.org

3, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241

Offset: 1

N. J. A. Sloane

Subsequence of A007401. - Reinhard Zumkeller, Jul 18 2011

Primes not dividing 10. Therefore primes p for which the decimal expansion of 1/p does not terminate (primes in A085837). - Jeppe Stig Nielsen, Dec 15 2019

John H. Conway, R. K. Guy, The Book of Numbers, Copernicus Press, p. 162.

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Subsequence of A065091, A007401, and A085837.

Prime@ Delete[Range[2, 53], {2}] (* Michael De Vlieger, Dec 15 2019 *)

a(n)=if(n>1,prime(n+2),3) \\ Charles R Greathouse IV, Aug 26 2011

A000365 Number of genus 0 rooted planar maps with 4 faces and n vertices.

Original entry on oeis.org

5, 93, 1030, 8885, 65954, 442610, 2762412, 16322085, 92400330, 505403910, 2687477780, 13957496098, 71053094420, 355548314180, 1752827693528, 8529176056965, 41026491589722, 195327793313790, 921451498774660, 4311086414580022, 20019238138410940

Offset: 3

N. J. A. Sloane

nonn

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
T. R. S. Walsh, Combinatorial Enumeration of Non-Planar Maps. Ph.D. Dissertation, Univ. of Toronto, 1971.

T. D. Noe, Table of n, a(n) for n = 3..200
W. T. Tutte, On the enumeration of planar maps, Bull. Amer. Math. Soc. 74 1968 64-74.
T. R. S. Walsh and A. B. Lehman, Counting rooted maps by genus, J. Comb. Thy B13 (1972), 122-141 and 192-218.
Notes

Column 4 of A269920.

Column 0 of A270408.

nn = 20; CoefficientList[Series[x^2 (1 - Sqrt[1 - 4 x]) (7 + 4 x - 2 Sqrt[1 - 4 x])/(2 (4 x - 1)^4), {x, 0, nn}], x] (* T. D. Noe, Jun 19 2012 *)

seq(n)={my(g=sqrt(1-4*x + O(x*x^n))); Vec((1-g)*(7+4*x-2*g)/(2*(1-4*x)^4))} \\ Andrew Howroyd, Mar 27 2021

G.f.: x^2*(1-sqrt(1-4*x))*(7+4*x-2*sqrt(1-4*x))/(2*(4*x-1)^4). - corrected for right offset by Vaclav Kotesovec, Aug 13 2013

a(n) ~ n^3*4^n/24 * (1-4/(sqrt(Pi*n))). - Vaclav Kotesovec, Aug 13 2013

More terms from Sean A. Irvine, Nov 14 2010

A005645 Number of sensed 3-connected planar maps with n edges.

Original entry on oeis.org

1, 0, 1, 2, 3, 4, 15, 32, 89, 266, 797, 2496, 8012, 26028, 85888, 286608, 965216, 3278776, 11221548, 38665192, 134050521, 467382224, 1638080277, 5768886048, 20407622631, 72494277840, 258527335373, 925322077852, 3323258053528, 11973883092034, 43273374700200, 156836969693756, 569967330200576, 2076647113454878, 7584534277720818, 27764845224462192, 101862027752012402, 374484866509396780, 1379489908513460150

Offset: 6

N. J. A. Sloane

nonn

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

N. J. A. Sloane, Table of n, a(n) for n = 6..50
T. R. S. Walsh, Number of sensed planar maps with n edges and m vertices
T. R. S. Walsh, Counting nonisomorphic three-connected planar maps, J. Combin. Theory Ser. B 32 (1982), no. 1, 33-44.

Cf. A002840 (unsensed), A239893.

a(n) = Sum_{k=4..n-2} A239893(k, n+2-k). - Andrew Howroyd, Mar 27 2021

More terms and b-file added by N. J. A. Sloane, May 08 2012

A006444 Number of achiral 2-connected planar maps with n edges.

Original entry on oeis.org

0, 1, 2, 3, 6, 14, 30, 77, 196, 525, 1414, 3960, 11056, 31636, 90818, 264657, 774146, 2289787, 6798562, 20354005, 61164374, 184985060, 561433922, 1712696708, 5241637812

Offset: 1

N. J. A. Sloane

nonn

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
T. R. S. Walsh, personal communication.

Timothy R. Walsh, Number of sensed planar maps with n edges and m vertices, p. 44.

Cf. A006402 (sensed), A006403 (unsensed), A006443 (connected), A006445 (3-connected).

A034854 Triangle giving number of labeled trees with n >= 3 nodes and diameter d >= 2.

Original entry on oeis.org

3, 4, 12, 5, 60, 60, 6, 210, 720, 360, 7, 630, 6090, 7560, 2520, 8, 1736, 47040, 112560, 80640, 20160, 9, 4536, 363384, 1496880, 1829520, 907200, 181440, 10, 11430, 2913120, 19207440, 36892800, 28274400, 10886400, 1814400

Offset: 0

N. J. A. Sloane

			Triangle begins:
  3;
  4,   12;
  5,   60,   60;
  6,  210,  720,  360;
  7,  630, 6090, 7560, 2520;
  ...

J. Riordan, Enumeration of trees by height and diameter, IBM J. Res. Dev. 4 (1960), 473-478.[broken link]
J. Riordan, Enumeration of trees by height and diameter, IBM J. Res. Dev. 4 (1960), 473-478.
J. Riordan, The enumeration of trees by height and diameter, IBM Journal 4 (1960), 473-478. (Annotated scanned copy)
Index entries for sequences related to trees

Cf. A001852, A034855, A356292, A355671.

Reference gives recurrence.
From Geoffrey Critzer, Aug 02 2022: (Start)
Sum_{d even} a(n,d) = A356292(n) and Sum_{d odd} a(n,d) = A355671(n).
Let G_k(x) be the e.g.f. counting the number of rooted labeled trees with height <= k. Then G_k(x) is defined recursively by G_0(x) = x, G_k(x) = x*exp(G_{k-1}(x)). Let H_k(x) be the e.g.f. counting rooted labeled trees of height k. Then H_0(x) = x, H_k(x) = G_k(x) - G_{k-1}(x) for k >= 1. The e.g.f. for column d = 2*m+1 is H_m(x)^2/2. The e.g.f. for column d = 2*m is G_{m-1}(x)*(exp(H_{m-1}(x)) - 1 - H_{m-1}(x)). (End)

More terms from Pab Ter (pabrlos(AT)yahoo.com), May 27 2004

Name corrected by Geoffrey Critzer, Aug 02 2022

cp's OEIS Frontend

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.

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A006443 Number of achiral planar maps with n edges.

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A006403 Number of unsensed 2-connected planar maps with n edges.

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A001853 Total height of trees with n nodes.

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A000184 Number of genus 0 rooted maps with 3 faces with n vertices.

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Magma

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SageMath

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A004139 Odd primes excluding 5.

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Mathematica

PARI

A000365 Number of genus 0 rooted planar maps with 4 faces and n vertices.

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Mathematica

PARI

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A005645 Number of sensed 3-connected planar maps with n edges.

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