A000168 -id:A000168 - cp's OEIS frontend

A380241 Array read by antidiagonals: T(n,k) is the number of rooted (2k)-regular planar maps with n vertices, n >= 0, k >= 0.

1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 5, 9, 1, 1, 1, 14, 100, 54, 1, 1, 1, 42, 1225, 3000, 378, 1, 1, 1, 132, 15876, 171500, 110000, 2916, 1, 1, 1, 429, 213444, 10001880, 30012500, 4550000, 24057, 1, 1, 1, 1430, 2944656, 591666768, 7981500240, 5987493750, 204000000, 208494, 1, 1

Offset: 0

Andrew Howroyd, Jan 22 2025

The zeroth column is included by convention only for consistency with the first row sequences.

The case for regular planar maps of odd valency is more complicated and without simple closed form formulas, so not presented in this sequence. See the references for additional information.

			Array begins:
====================================================================
n\k | 0  1      2          3               4                   5 ...
----+---------------------------------------------------------------
  0 | 1  1      1          1               1                   1 ...
  1 | 1  1      2          5              14                  42 ...
  2 | 1  1      9        100            1225               15876 ...
  3 | 1  1     54       3000          171500            10001880 ...
  4 | 1  1    378     110000        30012500          7981500240 ...
  5 | 1  1   2916    4550000      5987493750       7304332956480 ...
  6 | 1  1  24057  204000000   1302227368750    7310748066293952 ...
  7 | 1  1 208494 9690000000 301107909375000 7794097754539041792 ...
  ...

Andrew Howroyd, Table of n, a(n) for n = 0..1325 (first 51 antidiagonals)
E. A. Bender and E. R. Canfield, The number of degree restricted rooted maps on the sphere, SIAM J. Discrete Math. 7 (1994) 9-15.
Zhicheng Gao and Mizan Rahman, Enumeration of k-poles, Annals of Combinatorics 1 (1997), pp. 55-66.
W. T. Tutte, A Census of Slicings, Canad. J. Math. 14 (1962), 708-722.
W. T. Tutte, A Census of Planar Maps, Canad. J. Math. 15 (1963), 249-271.

Columns 0..3 are A000012 twice, A000168, A380242.
Rows 0..3 are A000012, A000108, A060150, A380243.
Cf. A269920.

T(n,k)=if(k==0, 1, 2*binomial(2*k-1,k)^n*(n*k)!/(n!*(n*k - n + 2)!))

T(n,k) = 2*binomial(2*k-1, k)^n*(n*k)!/(n!*(n*k - n + 2)!) for k > 0.

A380242 Number of rooted 6-regular planar maps with n vertices.

Original entry on oeis.org

1, 5, 100, 3000, 110000, 4550000, 204000000, 9690000000, 480700000000, 24667500000000, 1300650000000000, 70122000000000000, 3851316000000000000, 214878980000000000000, 12151776800000000000000, 695297229000000000000000, 40193385270000000000000000, 2344614140750000000000000000

Offset: 0

Andrew Howroyd, Jan 22 2025

nonn

Andrew Howroyd, Table of n, a(n) for n = 0..200

Column k=3 of A380241.

Cf. A000168, A380243.

PARI
```
a(n) = 2*10^n*(3*n)!/(n!*(2*n + 2)!)
```

a(n) = 2*10^n*(3*n)!/(n!*(2*n + 2)!).

A232546 Expansion of (1 - 12*x)^(3/2) in powers of x.

Original entry on oeis.org

1, -18, 54, 108, 486, 2916, 20412, 157464, 1299078, 11258676, 101328084, 939587688, 8926083036, 86514343272, 852784240824, 8527842408240, 86344404383430, 883760374277460, 9132190534200420, 95167038198509640, 999253901084351220, 10563541240034570040

Offset: 0

Michael Somos, Nov 25 2013

sign

From Ralf Steiner, Apr 04 2017: (Start)
By analytic continuation to the entire complex plane there exist regularized values for divergent sums such as:
Sum_{k>=0} a(k)^2/16^k = 2F1(-3/2,-3/2,1,9).
Sum_{k>=0} a(k) / 6^k = -i. (End)

			G.f. = 1 - 18*x + 54*x^2 + 108*x^3 + 486*x^4 + 2916*x^5 + 20412*x^6 + ...

G. C. Greubel, Table of n, a(n) for n = 0..930
R. Steiner, Sums of OEIS-A232546, ResearchGate, 2017. - _Ralf Steiner_, Apr 04 2017

Cf. A000168, A002421, A115903.

a[ n_] := SeriesCoefficient[ (1 - 12 x)^(3/2), {x, 0, n}];
Table[9/Sqrt[Pi] 12^n Gamma[-1/2 + n]/Gamma[2 + n], {n, -1, 20}] (* Ralf Steiner, Apr 01 2017 *)
Flatten[{1, -18, Table[4*3^(n+1)*(2*n-4)!/((n-2)!*n!), {n, 2, 25}]}] (* Vaclav Kotesovec, Apr 02 2017 *)

{a(n) = if( n<0, 0, polcoeff( (1 - 12 * x + x * O(x^n))^(3/2), n))};

0 = a(n+2)*(a(n+1) - 42*a(n)) + 18*a(n+1)*(a(n+1) + 8*a(n)) for all n in Z.
a(n+2) = 54 * A000168(n). a(n) = 3^n * A002421(n). Convolution inverse of A115903.
a(n) = 6*(2*n-5)*a(n-1)/n. - R. J. Mathar, Nov 23 2014
G.f.: 1F0(-3/2;;12x). - R. J. Mathar, Aug 09 2015
For n>=2, a(n) = 4*3^(n+1)*(2*n-4)! / ((n-2)!*n!). - Vaclav Kotesovec, Apr 02 2017
Sum_{k>=0} a(k) / 12^k = 0. - Ralf Steiner, Apr 04 2017

A379437 Number of rooted 2-connected simple planar maps with n edges.

Original entry on oeis.org

1, 1, 6, 16, 71, 267, 1162

Offset: 3

Andrew Howroyd, Jan 16 2025

A simple planar map is a planar map without loops or parallel edges.

Cf. A000139, A000168, A006406 (sensed), A006407 (unsensed).

A101486 Square array T(n,k), read by antidiagonals: number of labeled trees, with increments of labels along edges constrained to -1,0,1, with n nodes that have no label greater than k.

Original entry on oeis.org

1, 1, 2, 1, 3, 9, 1, 3, 17, 54, 1, 3, 18, 119, 378, 1, 3, 18, 134, 932, 2916, 1, 3, 18, 135, 1111, 7838, 24057, 1, 3, 18, 135, 1133, 9833, 69275, 208494, 1, 3, 18, 135, 1134, 10176, 90959, 635279, 1876446, 1, 3, 18, 135, 1134, 10205, 95635, 868827, 5994584, 222646205

Offset: 0

Ralf Stephan, Jan 21 2005

Rows converge to A005159.

First row is A000168.

			1,2,9,54,378,2916,24057,208494,1876446,17399772,
1,3,17,119,932,7838,69275,635279,5994584,57872666,
1,3,18,134,1111,9833,90959,868827,8504314,84866778,
1,3,18,135,1133,10176,95635,928442,9236144,93646430,
1,3,18,135,1134,10205,96191,937361,9365984,95427597,
1,3,18,135,1134,10206,96227,938179,9381050,95673739,
1,3,18,135,1134,10206,96228,938222,9382179,95697199,
1,3,18,135,1134,10206,96228,938223,9382229,95698688,

M. Bousquet-Mélou, Limit laws for embedded trees, arXiv:math/0501266 [math.CO], 2005.

Cf. A000168, A005159, A101487.

nmax = 9;
b[t_] = 2/(1 + Sqrt[1 - 12t]) + O[t]^(nmax+1);
c[t_] = (1 + Sqrt[1 - 12t] - t (8 + Sqrt[2] Sqrt[(1 + Sqrt[1 - 12t] - 2 (7 + 4 Sqrt[1 - 12t]) t + 24t^2)/t^2]))/(4t) + O[t]^(nmax+1) // Simplify[#, t > 0]&;
a[n_, t_] := a[n, t] = b[t] (1 - c[t]^(n + 1)) (1 - c[t]^(n + 4))/((1 - c[t]^(n+2)) (1 - c[t]^(n+3))) + O[t]^(nmax+1) // Simplify[#, t > 0]&;
T[n_, k_] := SeriesCoefficient[a[n, t], {t, 0, k}];
Table[T[n - k, k], {n, 0, nmax}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jul 25 2018 *)

G.f. of k-th row: A(t)=B(t)*(1-C(t)^(k+1))*(1-C(t)^(k+4))/[(1-C(t)^(k+2))*(1-C(t)^(k+3))], with B(t) the g.f. of A005159 and C(t) the g.f. of A101487.

A173794 Partial sums of A006384.

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

Jonathan Vos Post, Feb 24 2010

nonn

Partial sums of number of planar maps with n edges. The subsequence of primes in this partial sum begins: 3, 7, 17491, and no more known.

			a(21) = 1 + 2 + 4 + 14 + 57 + 312 + 2071 + 15030 + 117735 + 967850 + 8268816 + 72833730 + 658049140 + 6074058060 + 57106433817 + 545532037612 + 5284835906037 + 51833908183164 + 514019531037910 + 5147924676612282 + 52017438279806634 + 529867070532745464.

G. C. Greubel, Table of n, a(n) for n = 0..900

Cf. A000168, A006384.

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 *)

a(n) = Sum_{i=0..n} A006384(i).

A246323 Triangle read by rows: T(n,k) = number of normal planar lambda terms of size n with k free variables (n >= 1, 1 <= k <= n).

Original entry on oeis.org

1, 2, 1, 9, 6, 2, 54, 40, 20, 5, 378, 295, 175, 70, 14, 2916, 2346, 1526, 756, 252, 42, 24057, 19739, 13587, 7602, 3234, 924, 132, 208494, 173426, 123978, 74964, 36828, 13728, 3432, 429

Offset: 1

N. J. A. Sloane, Aug 28 2014

			Triangle begins:
1
2,1
9,6,2
54,40,20,5
378,295,175,70,14
2916,2346,1526,756,252,42
24057,19739,13587,7602,3234,924,132,
208494,173426,123978,74964,36828,13728,3432,429
...

Noam Zeilberger, Alain Giorgetti, A correspondence between rooted planar maps and normal planar lambda terms, arXiv:1408.5028 [cs.LO], 2014.

Cf. A000108, A000984, A000168, A246322.

A275607 a(n) = 212^nGamma(n+1/2)(n+1)/(sqrt(Pi)Gamma(n+3)).

Original entry on oeis.org

1, 4, 27, 216, 1890, 17496, 168399, 1667952, 16888014, 173997720, 1818276174, 19225409616, 205299909828, 2210922105840, 23984556773175, 261854925711840, 2874948871877910, 31722346066169880, 351589335566716170, 3912422681494285200, 43694647856506630620, 489597172255515289680

Offset: 0

Karol A. Penson, Nov 14 2016

nonn

In reference of K. Szymanski et al. the function g(x) from the Eq.(4.6) satisfies the equality g(x/4)/4 = W(x) where W(x) is the weight function of the integral representation, see below.

G. C. Greubel, Table of n, a(n) for n = 0..925
Simeon T. Stefanov, Counting fixed points free vector fields on B^2, arXiv:1807.03714 [math.GT], 2018.
K. Szymanski, B. Collins, T. Szarek and K. Zyczkowski, Convex set of quantum states with positive partial transpose analysed by hit and run algorithm, arXiv:1611.01194 [quant-ph], 2016.

Cf. A000108, A002293, A002894, A000168, A000139, A000257, A004987, A005568, A000888, A004981.

a := n -> (2^(2*n+1)*3^n*(n+1)*GAMMA(n+1/2))/(sqrt(Pi)*GAMMA(n+3)):
seq(a(n), n=0..21); # Peter Luschny, Nov 14 2016

g[z_] :=  E^z (BesselI[0,z] - (1-1/z) BesselI[1,z])
Table[CoefficientList[2/3 Series[g[6z], {z,0,21}],z]] Range[0, 21]! //Flatten (* Peter Luschny, Nov 14 2016 *)
Table[ 2*12^n*(n + 1)*Gamma[n + 1/2]/(Sqrt[Pi]*Gamma[n + 3]), {n,0,100}] (* G. C. Greubel, Jan 13 2017 *)

a(n)=2*12^n*gamma(n+1/2)*(n+1)\/(sqrt(Pi)*(n+2)!) \\ Charles R Greathouse IV, Nov 14 2016

a(n)=2*3^n*binomial(2*n+1,n-1)*(n+1)/(2*n+1)/n \\ Charles R Greathouse IV, Nov 14 2016

O.g.f: (1/54)*(1-(6*z+1)*sqrt(1-12*z))/z^2;
E.g.f.(in Maple notation): (1/9)*exp(6*z)*(6*z*(BesselI(0,6*z)-BesselI(1,6*z))+ BesselI(1,6*z))/z;
Recurrence: (-12*n^2-54*n-54)*a(n+1)+(n^2+6*n+8)*a(n+2)=0, n=0,1..., for the initial values a(0)=1, a(1)=4.
Integral representation as the n-th Hausdorff moment of the positive function W(x) on the segment x=(0,12), i.e., a(n) = Integral_{x=0..12} x^n*W(x) dx, where W(x) = (1/27)*sqrt(12-x)*(3+(1/2)*x)/(Pi*sqrt(x)). This representation is unique.
a(n) ~ 2^(2*n+1)*3^n/(sqrt(Pi)*n^(3/2)). - Ilya Gutkovskiy, Nov 14 2016
a(n) = 2*3^n*binomial(2n+1, n-1)*(n+1)/(2n^2+n). - Charles R Greathouse IV, Nov 14 2016

A318106 Triangle read by rows: T(n,k) is the number of rooted maps with n edges whose core comprises k edges, 1 <= k <= n.

Original entry on oeis.org

2, 8, 1, 44, 8, 2, 288, 60, 24, 6, 2106, 464, 228, 96, 22, 16632, 3742, 2048, 1104, 440, 91, 138996, 31392, 18246, 11328, 5940, 2184, 408, 1213056, 272592, 163896, 111048, 68640, 33852, 11424, 1938, 10955412, 2438208, 1493012, 1070016, 736230, 435344, 199920, 62016, 9614, 101721744, 22369365, 13816224, 10270752, 7602408, 5079438, 2833152, 1209312, 346104, 49335

Offset: 1

Gheorghe Coserea, Sep 22 2018

			A(x;t) = 2*t*x + (8*t + t^2)*x^2 + (44*t + 8*t^2 + 2*t^3)*x^3 + ...
Triangle starts:
n\k [1]       [2]      [3]      [4]      [5]     [6]     [7]     [8]    [9]
[1] 2;
[2] 8,        1;
[3] 44,       8,       2;
[4] 288,      60,      24,      6;
[5] 2106,     464,     228,     96,      22;
[6] 16632,    3742,    2048,    1104,    440,    91;
[7] 138996,   31392,   18246,   11328,   5940,   2184,   408;
[8] 1213056,  272592,  163896,  111048,  68640,  33852,  11424,  1938;
[9] 10955412, 2438208, 1493012, 1070016, 736230, 435344, 199920, 62016, 9614;
[10]...

Gheorghe Coserea, Rows n=1..202, flattened
Cyril Banderier, Philippe Flajolet, Gilles Schaeffer, Michele Soria, Random maps, coalescing saddles, singularity analysis, and Airy phenomena, Random Structures and Algorithms 19(3-4), 2001.

Row sums give A000168 for n>=1.

Main diagonal give A000139(n-1) for n>=1.

A000139[x_] = 2/(3x) (HypergeometricPFQ[{-2/3, -1/3}, {1/2}, (27/4) x]-1);
A000168[x_] = HypergeometricPFQ[{1/2, 1}, {3}, 12 x];
h[x_] = x A000168[x]^2;
A[x_, t_] := t h[x] A000139[t h[x]];
Rest[CoefficientList[#, t]]& /@ Rest[CoefficientList[A[x, t] + O[x]^11, x]] // Flatten (* Jean-François Alcover, Aug 29 2019 *)

seq(N) = {
  my(x='x + O('x^(N+3)),  m=(-1 + 18*x + (1-12*x)^(3/2))/(54*x^2),
     h=x*m^2, c=subst(m, 'x, serreverse(h)));
  apply(Vecrev, Vec((subst(c, 'x, 't*h) - 1)/'t));
};
seq(10)

G.f.: A(x;t) = t*h*A000139(t*h), where h=x*A000168(x)^2 (see eqn. (15) in Banderier link).

A361137 Number of rooted maps of genus 1/2 with n edges.

Original entry on oeis.org

1, 10, 98, 983, 10062, 105024, 1112757, 11934910, 129307100, 1412855500, 15548498902, 172168201088, 1916619748084, 21436209373224, 240741065193282, 2713584138389838

Offset: 1

R. J. Mathar, Mar 02 2023

Valentin Bonzom, Guillaume Chapuy, and Maciej Dolega, Enumeration of non-oriented maps via integrability, Alg. Combin. 5 (6) (2022) pp. 1363-1390, A.1.

Cf. A000168 (genus 0).

cp's OEIS Frontend

A380241 Array read by antidiagonals: T(n,k) is the number of rooted (2k)-regular planar maps with n vertices, n >= 0, k >= 0.

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A380242 Number of rooted 6-regular planar maps with n vertices.

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A232546 Expansion of (1 - 12*x)^(3/2) in powers of x.

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Mathematica

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A379437 Number of rooted 2-connected simple planar maps with n edges.

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A101486 Square array T(n,k), read by antidiagonals: number of labeled trees, with increments of labels along edges constrained to -1,0,1, with n nodes that have no label greater than k.

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A173794 Partial sums of A006384.

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A246323 Triangle read by rows: T(n,k) = number of normal planar lambda terms of size n with k free variables (n >= 1, 1 <= k <= n).

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A275607 a(n) = 2*12^n*Gamma(n+1/2)*(n+1)/(sqrt(Pi)*Gamma(n+3)).

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A275607 a(n) = 212^nGamma(n+1/2)(n+1)/(sqrt(Pi)Gamma(n+3)).