A000168 -id:A000168 - cp's OEIS frontend

A238359 Number of genus-9 rooted maps with n edges.

11665426077721040625, 3498878057690404966500, 540996834819906946713375, 57494374008560749302297480, 4724172886681078698955547790, 320061005837218582787265273000, 18618409220753939214291224549409, 956146512935178711199035220384800, 44232688287025023758415781081779828, 1871678026675570344184400604255444240

Offset: 18

Joerg Arndt, Feb 26 2014

nonn

Andrew Howroyd, Table of n, a(n) for n = 18..200
Sean R. Carrell and Guillaume Chapuy, Simple recurrence formulas to count maps on orientable surfaces, arXiv:1402.6300 [math.CO], (19-March-2014).
Gheorghe Coserea, The g.f. as a rational function of y=A005159(x)
Steven R. Finch, An exceptional convolutional recurrence, arXiv:2408.12440 [math.CO], 22 Aug 2024.

Column g=9 of A269919.

Rooted maps with n edges of genus g for 0 <= g <= 10: A000168, A006300, A006301, A104742, A215402, A238355, A238356, A238357, A238358, this sequence, A238360.

T[0, 0] = 1; T[n_, g_] /; g < 0 || g > n/2 = 0; T[n_, g_] := T[n, g] = ((4 n - 2)/3 T[n - 1, g] + (2 n - 3) (2 n - 2) (2 n - 1)/12 T[n - 2, g - 1] + 1/2 Sum[(2 k - 1) (2 (n - k) - 1) T[k - 1, i] T[n - k - 1, g - i], {k, 1, n - 1}, {i, 0, g}])/((n + 1)/6);
a[n_] := T[n, 9];
Table[a[n], {n, 18, 30}] (* Jean-François Alcover, Jul 20 2018 *)

PARI
```
\\ see A238396
```

A238360 Number of genus-10 rooted maps with n edges.

Original entry on oeis.org

15230046989184655753125, 5199629454143883380553750, 909887917857275652461097750, 108861830345440643086946970900, 10021124647635764856828690342402, 757187906770815991999545249667404, 48918614114003431712044170834572688, 2779227352989564224315657269511192976, 141720718575991991799057452443438430580

Offset: 20

Joerg Arndt, Feb 26 2014

nonn

Andrew Howroyd, Table of n, a(n) for n = 20..200
Sean R. Carrell and Guillaume Chapuy, Simple recurrence formulas to count maps on orientable surfaces, arXiv:1402.6300 [math.CO], (19-March-2014).
Gheorghe Coserea, The g.f. as a rational function of y=A005159(x)
Steven R. Finch, An exceptional convolutional recurrence, arXiv:2408.12440 [math.CO], 22 Aug 2024.

Column g=10 of A269919.

Rooted maps with n edges of genus g for 0 <= g <= 10: A000168, A006300, A006301, A104742, A215402, A238355, A238356, A238357, A238358, A238359, this sequence.

T[0, 0] = 1; T[n_, g_] /; g < 0 || g > n/2 = 0; T[n_, g_] := T[n, g] = ((4 n - 2)/3 T[n - 1, g] + (2 n - 3) (2 n - 2) (2 n - 1)/12 T[n - 2, g - 1] + 1/2 Sum[(2 k - 1) (2 (n - k) - 1) T[k - 1, i] T[n - k - 1, g - i], {k, 1, n - 1}, {i, 0, g}])/((n + 1)/6);
a[n_] := T[n, 10];
Table[a[n], {n, 20, 30}] (* Jean-François Alcover, Jul 20 2018 *)

PARI
```
\\ see A238396
```

A267827 Number of closed indecomposable linear lambda terms with 2n+1 applications and abstractions.

Original entry on oeis.org

1, 2, 20, 352, 8624, 266784, 9896448, 426577920, 20918138624, 1149216540160, 69911382901760, 4665553152081920, 338942971881472000, 26631920159494995968, 2250690001888540950528, 203595258621775065120768, 19629810220331494121865216

Offset: 0

Noam Zeilberger, Jan 21 2016

nonn

A linear lambda term is indecomposable if it has no closed proper subterm.
Equivalently, number of closed bridgeless rooted trivalent maps (on compact oriented surfaces of arbitrary genus) with 2n+1 trivalent vertices (and 1 univalent vertex).
The September 2018 talk by Noam Zeilberger (see link to video) connects three topics (planar maps, Tamari lattices, lambda calculus) and eight sequences: A000168, A000260, A000309, A000698, A000699, A002005, A062980, A267827. - N. J. A. Sloane, Sep 17 2018

			A(x) = 1 + 2*x + 20*x^2 + 352*x^3 + 8624*x^4 + 266784*x^5 + ...

Gheorghe Coserea, Table of n, a(n) for n = 0..303
Lawrence Dresner, Protection of a test magnet wound with a Ag/BSCCO high-temperature superconductor, Oak Ridge National Lab technical report (ORNL/HTSPC-3), 1992. See Eq. (25).
Noam Zeilberger, Linear lambda terms as invariants of rooted trivalent maps, arXiv:1512.06751 [cs.LO], 2015.
Noam Zeilberger, A theory of linear typings as flows on 3-valent graphs, arXiv:1804.10540 [cs.LO], 2018.
Noam Zeilberger, A Sequent Calculus for a Semi-Associative Law, arXiv preprint 1803.10030, March 2018 (A revised version of a 2017 conference paper)
Noam Zeilberger, A proof-theoretic analysis of the rotation lattice of binary trees, Part 1 (video), Rutgers Experimental Math Seminar, Sep 13 2018. Part 2 is vimeo.com/289910554.
Noam Zeilberger, From Lambda Calculus to the Four Color Theorem, via Experimental Mathematics (slides), Rutgers Experimental Math Seminar, Jun 18 2020. For the video see http://noamz.org/videos/expmath.2020.06.18.mp4.
Noam Zeilberger, From Lambda Calculus to the Four Color Theorem, via Experimental Mathematics (slides), Rutgers Experimental Math Seminar, Jun 18 2020. [Local copy]

Sequences mentioned in the Noam Zeilberger 2018 video: A000168, A000260, A000309, A000698, A000699, A002005, A062980, A267827.

Cf. A000309, A062980.

a[0] = 1; a[1] = 2; a[n_] := a[n] = (6n-2) a[n-1] + Sum[(6k+2) a[k] a[n-1-k], {k, 1, n-2}];
Table[a[n], {n, 0, 16}] (* Jean-François Alcover, Oct 16 2018, after Gheorghe Coserea *)

seq(N) = {
  my(a = vector(N)); a[1] = 2;
  for(n=2, N,
    a[n] = (6*n-2)*a[n-1] + sum(k=1, n-2, (6*k+2)*a[k]*a[n-1-k]));
  concat(1,a);
};
seq(16)
\\ test 1: y = x^2*subst(Ser(seq(201)),'x,-'x^6); 0 == x^5*y*y' + y - x^2
\\ test 2: y = Ser(seq(201)); 0 == 6*y*y'*x^2 + 2*y^2*x - y + 1
\\ Gheorghe Coserea, Nov 10 2017
F(N) = {
  my(x='x+O('x^N), t='t, F0=x, F1=0, n=1);
  while(n++,
    F1 = t + x*(F0 - subst(F0,t,0))^2 + x*deriv(F0,t);
    if (F1 == F0, break()); F0 = F1;);
  F0;
};
seq(N) = my(v=Vec(subst(F(2*N+2),'t,0))); vector((#v+1)\2, n, v[2*n-1]);
seq(16) \\ Gheorghe Coserea, Apr 01 2017

The o.g.f. f(z) = z + 2*z^3 + 20*z^5 + 352*z^7 + ... can be defined using a catalytic variable as f(z) = F(z,0), where F(z,x) satisfies the functional-differential equation F(z,x) = x + z*(F(z,x) - F(z,0))^2 + z*(d/dx)F(z,x).
From Gheorghe Coserea, Nov 10 2017: (Start)
0 = x^5*y*y' + y - x^2, where y(x) = x^2*A(-x^6).
0 = 6*y*y'*x^2 + 2*y^2*x - y + 1, where y(x) = A(x).
a(n) = (6*n-2)*a(n-1) + Sum_{k=1..n-2} (6*k+2)*a(k)*a(n-1-k), for n >= 2.
(End)
a(n) = A291843(3*n+1, 2*n), n >= 1. - Danny Rorabaugh, Nov 10 2017

A269920 Triangle read by rows: T(n,f) is the number of rooted maps with n edges and f faces on an orientable surface of genus 0.

Original entry on oeis.org

1, 1, 1, 2, 5, 2, 5, 22, 22, 5, 14, 93, 164, 93, 14, 42, 386, 1030, 1030, 386, 42, 132, 1586, 5868, 8885, 5868, 1586, 132, 429, 6476, 31388, 65954, 65954, 31388, 6476, 429, 1430, 26333, 160648, 442610, 614404, 442610, 160648, 26333, 1430

Offset: 0

Gheorghe Coserea, Mar 14 2016

Row n contains n+1 terms.

			Triangle starts:
n\f    [1]     [2]     [3]     [4]     [5]     [6]     [7]     [8]
[0]    1;
[1]    1,      1;
[2]    2,      5,      2;
[3]    5,      22,     22,     5;
[4]    14,     93,     164,    93,     14;
[5]    42,     386,    1030,   1030,   386,    42;
[6]    132,    1586,   5868,   8885,   5868,   1586,   132;
[7]    429,    6476,   31388,  65954,  65954,  31388,  6476,   429;
[8]    ...

Gheorghe Coserea, Rows n = 0..200, flattened
Sean R. Carrell, Guillaume Chapuy, Simple recurrence formulas to count maps on orientable surfaces, arXiv:1402.6300 [math.CO], 2014.

Columns k=1-6 give: A000108, A000346, A000184, A000365, A000473, A000502.
Row sums give A000168 (column 0 of A269919).
Cf. A006294 (row maxima).

Q[0, 1, 0] = 1; Q[n_, f_, g_] /; n<0 || f<0 || g<0 = 0;
Q[n_, f_, g_] := Q[n, f, g] = 6/(n+1) ((2n-1)/3 Q[n-1, f, g] + (2n-1)/3 Q[n - 1, f-1, g] + (2n-3) (2n-2) (2n-1)/12 Q[n-2, f, g-1] + 1/2 Sum[l = n-k; Sum[v = f-u; Sum[j = g-i; Boole[l >= 1 && v >= 1 && j >= 0] (2k-1) (2l-1) Q[k-1, u, i] Q[l-1, v, j], {i, 0, g}], {u, 1, f}], {k, 1, n}]);
Table[Q[n, f, 0], {n, 0, 8}, {f, 1, n+1}] // Flatten (* Jean-François Alcover, Aug 10 2018 *)

N = 8; G = 0; gmax(n) = min(n\2, G);
Q = matrix(N + 1, N + 1);
Qget(n, g) = { if (g < 0 || g > n/2, 0, Q[n+1, g+1]) };
Qset(n, g, v) = { Q[n+1, g+1] = v };
Quadric({x=1}) = {
  Qset(0, 0, x);
  for (n = 1, length(Q)-1, for (g = 0, gmax(n),
    my(t1 = (1+x)*(2*n-1)/3 * Qget(n-1, g),
       t2 = (2*n-3)*(2*n-2)*(2*n-1)/12 * Qget(n-2, g-1),
       t3 = 1/2 * sum(k = 1, n-1, sum(i = 0, g,
       (2*k-1) * (2*(n-k)-1) * Qget(k-1, i) * Qget(n-k-1, g-i))));
    Qset(n, g, (t1 + t2 + t3) * 6/(n+1))));
};
Quadric('x);
concat(apply(p->Vecrev(p/'x), vector(N+1 - 2*G, n, Qget(n-1 + 2*G, G))))

A002005 Number of rooted planar cubic maps with 2n vertices.

Original entry on oeis.org

1, 4, 32, 336, 4096, 54912, 786432, 11824384, 184549376, 2966845440, 48855252992, 820675092480, 14018773254144, 242919827374080, 4261707069259776, 75576645116559360, 1353050213048123392, 24428493151359467520, 444370175232646840320, 8138178004138611179520

Offset: 0

N. J. A. Sloane

nonn

Equivalently, number of rooted planar triangulations with 2n faces.

The September 2018 talk by Noam Zeilberger (see link to video) connects three topics (planar maps, Tamari lattices, lambda calculus) and eight sequences: A000168, A000260, A000309, A000698, A000699, A002005, A062980, A267827. - N. J. A. Sloane, Sep 17 2018

R. C. Mullin, E. Nemeth and P. J. Schellenberg, The enumeration of almost cubic maps, pp. 281-295 in Proceedings of the Louisiana Conference on Combinatorics, Graph Theory and Computer Science. Vol. 1, edited R. C. Mullin et al., 1970.
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).

Gheorghe Coserea, Table of n, a(n) for n = 0..1000
Valentin Bonzom, Guillaume Chapuy, Maciej Dolega, Enumeration of non-oriented maps via integrability, Alg. Combin. 5 (6) (2022) p 1363-1390, A.3
Mireille Bousquet-Mélou, Counting planar maps, coloured or uncoloured, 23rd British Combinatorial Conference, Jul 2011, Exeter, United Kingdom. 392, pp.1-50, 2011, London Math. Soc. Lecture Note Ser., hal-00653963. See p.13.
Evgeniy Krasko and Alexander Omelchenko, Enumeration of r-regular maps on the torus. Part I: Rooted maps on the torus, the projective plane and the Klein bottle. Sensed maps on the torus, Discrete Mathematics (2019) Vol. 342, Issue 2, 584-599. Also arXiv:1709.03225 [math.CO].
Maxim Krikun, Explicit enumeration of triangulations with multiple boundaries, arXiv:0706.0681 [math.CO], 2007. [Comment from _Gheorghe Coserea_, Dec 26 2015: the formula in the paper for almost trivalent maps is 2 * 4^(k-1) * (3k)!!/ ((k+1)!*(k+2)!!); however, the exponent of 4 should be k not (k-1) i.e. 2 * 4^k * (3k)!! / ((k+1)!*(k+2)!!)]
Noam Zeilberger, A theory of linear typings as flows on 3-valent graphs, arXiv:1804.10540 [cs.LO], 2018.
Noam Zeilberger, A Sequent Calculus for a Semi-Associative Law, arXiv preprint 1803.10030 [math.LO], March 2018 (A revised version of a 2017 conference paper).
Noam Zeilberger, A proof-theoretic analysis of the rotation lattice of binary trees, Part 1 (video), Part 2, Rutgers Experimental Math Seminar, Sep 13 2018.
Jian Zhou, Fat and Thin Emergent Geometries of Hermitian One-Matrix Models, arXiv:1810.03883 [math-ph], 2018.

Sequences mentioned in the Noam Zeilberger 2018 video: A000168, A000260, A000309, A000698, A000699, A002005, A062980, A267827.
Column k=0 of A266240.
Cf. A000217, A358367.

seq(2*8^n*binomial(n*3/2, n)/((n + 2)*(n + 1)), n = 0..19); # Peter Luschny, Nov 14 2022

Table[2^(2 n + 1) (3 n)!!/((n + 2)! n!!), {n, 0, 20}] (* Vincenzo Librandi, Dec 28 2015 *)
CoefficientList[Series[(-1 + 96 z + Hypergeometric2F1[-2/3,-1/3,1/2,432z^2]- 96 z Hypergeometric2F1[-1/6,1/6,3/2,432z^2])/(192 z^2), {z, 0, 10}], z] (* Benedict W. J. Irwin, Aug 07 2016 *)

factorial2(n) = my(x = (2^(n\2)*(n\2)!)); if (n%2, n!/x, x);
a(n) = 2^(2*n+1)*factorial2(3*n)/((n+2)!*factorial2(n));
vector(20, i, a(i-1))
\\ test: y = Ser(vector(201, n, a(n-1))); x*(1-432*x^2)*y' == 64*x^2*y^2 + (288*x^2 - 64*x - 1)*y + 72*x + 1
\\ Gheorghe Coserea, Jun 13 2017

a(n) = 2^(2*n+1)*(3*n)!!/((n+2)!*n!!). - Sean A. Irvine, May 19 2013
a(n) ~ sqrt(6/Pi) * n^(-5/2) * (12*sqrt(3))^n. - Gheorghe Coserea, Feb 25 2016
G.f.: (96*x - 1 + 2F1(-2/3, -1/3; 1/2; 432*x^2) - 96*x*2F1(-1/6, 1/6; 3/2; 432*x^2))/(192*x^2). - Benedict W. J. Irwin, Aug 07 2016
From Gheorghe Coserea, Jun 13 2017: (Start)
G.f. y(x) satisfies:
x*(1-432*x^2)*deriv(y,x) = 64*x^2*y^2 + (288*x^2 - 64*x - 1)*y + 72*x + 1.
0 = 64*x^3*y^3 + x*(1-96*x)*y^2 + (30*x-1)*y - 27*x + 1.
(End).
D-finite with recurrence (n+2)*(n+1)*a(n) -48*(3*n-2)*(3*n-4)*a(n-2)=0. - R. J. Mathar, Feb 08 2021
From Karol A. Penson and Katarzyna Gorska (katarzyna.gorska@ifj.edu.pl), Nov 02 2022: (Start)
a(n) = Integral_{x=0..12*sqrt(3)} x^n*W(x), where
W(x) = (T1(x) + T2(x)) / T3(x), and
T1(x) = -x^(2/3) * (108 + sqrt(3) * sqrt(432 - x^2));
T2(x) = 3^(1/6)*(36+sqrt(3)*sqrt(432-x^2))^(2/3) * (-432+x^2+36*sqrt(3)* sqrt(432-x^2)) / sqrt(432-x^2);
T3(x) = (128*3^(5/6)*Pi*x^(1/3)*(36+sqrt(3)*sqrt(432-x^2))^(1/3)).
This integral representation is unique as W(x) is the solution of the Hausdorff power moment problem. Using only the definition of a(n), W(x) can be proven to be positive. W(x) is singular at x = 0, with the singularity x^(-1/3), and for x > 0 is monotonically decreasing to zero at x = 12*sqrt(3). (End)
a(n) = 2^(3*n + 1)*binomial(n*3/2, n)/((n + 1)*(n + 2)) = A358367(n) / A000217(n + 1). - Peter Luschny, Nov 14 2022

More terms from Sean A. Irvine, May 19 2013

A238396 Triangle T(n,k) read by rows: T(n,k) is the number of rooted genus-k maps with n edges, n>=0, 0<=k<=n.

Original entry on oeis.org

1, 2, 0, 9, 1, 0, 54, 20, 0, 0, 378, 307, 21, 0, 0, 2916, 4280, 966, 0, 0, 0, 24057, 56914, 27954, 1485, 0, 0, 0, 208494, 736568, 650076, 113256, 0, 0, 0, 0, 1876446, 9370183, 13271982, 5008230, 225225, 0, 0, 0, 0, 17399772, 117822512, 248371380, 167808024, 24635754, 0, 0, 0, 0, 0, 165297834, 1469283166, 4366441128, 4721384790, 1495900107, 59520825, 0

Offset: 0

Joerg Arndt, Feb 26 2014

			Triangle starts:
00: 1,
01: 2, 0,
02: 9, 1, 0,
03: 54, 20, 0, 0,
04: 378, 307, 21, 0, 0,
05: 2916, 4280, 966, 0, 0, 0,
06: 24057, 56914, 27954, 1485, 0, 0, 0,
07: 208494, 736568, 650076, 113256, 0, 0, 0, 0,
08: 1876446, 9370183, 13271982, 5008230, 225225, 0, 0, 0, 0,
09: 17399772, 117822512, 248371380, 167808024, 24635754, 0, ...,
10: 165297834, 1469283166, 4366441128, 4721384790, 1495900107, 59520825, 0, ...,
11: 1602117468, 18210135416, 73231116024, 117593590752, 66519597474, 8608033980, 0, ...,
12: 15792300756, 224636864830, 1183803697278, 2675326679856, 2416610807964, 672868675017, 24325703325, 0, ...,
...

David M. Jackson and Terry I. Visentin, An Atlas of the Smaller Maps in Orientable and Nonorientable Surfaces, Chapman & Hall/CRC, circa 2000. See page 227.

Joerg Arndt, Table of n, a(n) for n = 0..1325 (rows 0..50, flattened)
Sean R. Carrell, Guillaume Chapuy, Simple recurrence formulas to count maps on orientable surfaces, arXiv:1402.6300 [math.CO], (19-March-2014).

Columns k for 0<=k<=10 are: A000168, A006300, A006301, A104742, A215402, A238355, A238356, A238357, A238358, A238359, A238360.
Sum of row n is A000698(n+1).
See A267180 for nonorientable analog.
Cf. A269418, A269419.
The triangle without the zeros is A269919.

T[0, 0] = 1; T[n_, g_] /; g < 0 || g > n/2 = 0; T[n_, g_] := T[n, g] = ((4n - 2)/3 T[n-1, g] + (2n-3)(2n-2)(2n-1)/12 T[n-2, g-1] + 1/2 Sum[(2k-1)(2(n - k)-1) T[k-1, i] T[n-k-1, g-i] , {k, 1, n-1}, {i, 0, g}])/((n+1)/6);
Table[T[n, g], {n, 0, 10}, {g, 0, n}] // Flatten (* Jean-François Alcover, Jul 19 2018, after Gheorghe Coserea *)

N=20;
MEM=matrix(N+1,N+1, r,c, -1);  \\ for memoization
Q(n,g)=
{
    if (n<0,  return( (g<=0) ) ); \\ not given in paper
    if (g<0,  return( 0 ) ); \\ not given in paper
    if (n<=0, return( g==0 ) );  \\ as in paper
    my( m = MEM[n+1,g+1] );
    if ( m != -1,  return(m) );  \\ memoized value
    my( t=0 );
    t += (4*n-2)/3 * Q(n-1, g);
    t += (2*n-3)*(2*n-2)*(2*n-1)/12 * Q(n-2, g-1);
    my(l, j);
    t += 1/2*
        sum(k=1, n-1, l=n-k;  \\ l+k == n, both >= 1
            sum(i=0, g, j=g-i;  \\ i+j == g, both >= 0
                (2*k-1)*(2*l-1) * Q(k-1, i) * Q(l-1, j)
            );
        );
    t *= 6/(n+1);
    MEM[n+1, g+1] = t;  \\ memoize
    return(t);
}
for (n=0, N, for (g=0, n, print1(Q(n, g),", "); );  print(); ); /* print triangle */

From Gheorghe Coserea, Mar 11 2016: (Start)
(n+1)/6 * T(n, g) = (4*n-2)/3 * T(n-1, g) + (2*n-3)*(2*n-2)*(2*n-1)/12 * T(n-2, g-1) + 1/2 * Sum_{k=1..n-1} Sum_{i=0..g} (2*k-1) * (2*(n-k)-1) * T(k-1, i) * T(n-k-1, g-i) for all n >= 1 and 0 <= g <= n/2, with the initial conditions T(0,0) = 1 and T(n,g) = 0 for g < 0 or g > n/2.
For column g, as n goes to infinity we have T(n,g) ~ t(g) * n^(5*(g-1)/2) * 12^n, where t(g) = (A269418(g)/A269419(g)) / (2^(g-2) * gamma((5*g-1)/2)) and gamma is the Gamma function.
(End)

A089574 Column 4 of an array closely related to A083480. (Both arrays have shape sequence A083479).

Original entry on oeis.org

5, 32, 113, 299, 664, 1309, 2366, 4002, 6423, 9878, 14663, 21125, 29666, 40747, 54892, 72692, 94809, 121980, 155021, 194831, 242396, 298793, 365194, 442870, 533195, 637650, 757827, 895433, 1052294, 1230359, 1431704, 1658536, 1913197

Offset: 1

Alford Arnold, Dec 29 2003; extended May 04 2005

The diagonals are finite and sum to A047970.
Values appear to be a transformation of A006468 (rooted planar maps). Also known as well-labeled trees (cf. A000168).
First differences of the conjectured polynomial formula for A006468. [From R. J. Mathar, Jun 26 2010]

			The array begins
1
2
4
7 1
11 5
16 14 2
22 30 12
29 55 39 5
37 91 95 32 1

Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

Cf. A105552, A006468.
Cf. A006011, A034261.
Cf. A000124 (column 1), A000330 (column 2), A086602 (column 3), A107600 (column 5), A107601 (column 6), A109125 (column 7), A109126 (column 8), A109820 (column 9), A108538 (column 10), A109821 (column 11), A110553 (column 12), A110624 (column 13).

Row sums are powers of 2.
a(n) = A000330(n) + A006011(n+1) + A034263(n-1).
a(n)= +6*a(n-1) -15*a(n-2) +20*a(n-3) -15*a(n-4) +6*a(n-5) -a(n-6). G.f.: x*(5+2*x-4*x^2+x^3)/(x-1)^6. a(n) = n*(n+1)*(4*n^3+51*n^2+159*n+86)/120. [From R. J. Mathar, Jun 26 2010]

Extended beyond a(8) by R. J. Mathar, Jun 26 2010

A379433 Number of rooted planar maps with n edges and without faces of degree 1.

Original entry on oeis.org

1, 1, 3, 16, 96, 624, 4304, 31056, 232128, 1784752, 14043312, 112648848, 918456608, 7593649392, 63546379152, 537427956688, 4587713701248, 39488179213872, 342414691125104, 2989022121125136, 26249475365186016, 231786459869636464, 2056950693208881744

Offset: 0

Andrew Howroyd, Jan 14 2025

nonn

Cf. A000168, A006388 (sensed), A006389 (unsensed), A379434, A379435.

seq(n)={my(z=x/(1 + x)^2, g=(-1 + 18*z + sqrt(1-12*z + O(x^(n+3)))^3) / (54*z^2)); Vec(x + g*(1-x)/(1+x))}

G.f.: x + G(x/(1 + x)^2)*(1 - x)/(1 + x) where G(x) is the g.f. of A000168.

A379434 Number of rooted planar maps with n edges and without faces of degree 1 or 2.

Original entry on oeis.org

1, 0, 2, 9, 47, 278, 1720, 11175, 75149, 519852

Offset: 0

Andrew Howroyd, Jan 14 2025

Cf. A000168, A006392 (sensed), A006393 (unsensed), A379433.

A379435 Number of rooted planar maps with n edges and without faces or vertices of degree 1.

Original entry on oeis.org

1, 0, 1, 2, 10, 52, 281, 1570, 9022, 53084, 318634, 1945396, 12052532, 75624616, 479814937, 3074251682, 19869323638, 129420288076, 848897059790, 5603350613308, 37198680816844, 248241480270680, 1664546969372554, 11210468046615412, 75806810042727980, 514537522249147672

Offset: 0

Andrew Howroyd, Jan 14 2025

nonn

Cf. A000168, A006396 (sensed), A006397 (unsensed), A379433.

seq(n)={my(z=x/(1 + 2*x)^2, g=(-1 + 18*z + sqrt(1-12*z + O(x^(n+3)))^3) / (54*z^2)); Vec(2*x + g*(1-2*x)/(1+2*x))}

G.f.: 2*x + G(x/(1 + 2*x)^2)*(1 - 2*x)/(1 + 2*x) where G(x) is the g.f. of A000168.

cp's OEIS Frontend

A238359 Number of genus-9 rooted maps with n edges.

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A238360 Number of genus-10 rooted maps with n edges.

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A267827 Number of closed indecomposable linear lambda terms with 2n+1 applications and abstractions.

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A269920 Triangle read by rows: T(n,f) is the number of rooted maps with n edges and f faces on an orientable surface of genus 0.

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A002005 Number of rooted planar cubic maps with 2n vertices.

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A238396 Triangle T(n,k) read by rows: T(n,k) is the number of rooted genus-k maps with n edges, n>=0, 0<=k<=n.

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A089574 Column 4 of an array closely related to A083480. (Both arrays have shape sequence A083479).

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A379433 Number of rooted planar maps with n edges and without faces of degree 1.

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