cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-4 of 4 results.

A002293 Number of dissections of a polygon: binomial(4*n, n)/(3*n + 1).

Original entry on oeis.org

1, 1, 4, 22, 140, 969, 7084, 53820, 420732, 3362260, 27343888, 225568798, 1882933364, 15875338990, 134993766600, 1156393243320, 9969937491420, 86445222719724, 753310723010608, 6594154339031800, 57956002331347120, 511238042454541545
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Comments

The number of rooted loopless n-edge maps in the plane (planar with a distinguished outside face). - Valery A. Liskovets, Mar 17 2005
Number of lattice paths from (1,0) to (3*n+1,n) which, starting from (1,0), only utilize the steps +(1,0) and +(0,1) and additionally, the paths lie completely below the line y = (1/3)*x (i.e., if (a,b) is in the path, then b < a/3). - Joseph Cooper (jecooper(AT)mit.edu), Feb 07 2006
Number of length-n restricted growth strings (RGS) [s(0), s(1), ..., s(n-1)] where s(0) = 0 and s(k) <= s(k-1) + 3, see fxtbook link below. - Joerg Arndt, Apr 08 2011
From Wolfdieter Lang, Sep 14 2007: (Start)
a(n), n >= 1, enumerates quartic trees (rooted, ordered, incomplete) with n vertices (including the root).
Pfaff-Fuss-Catalan sequence C^{m}_n for m = 4. See the Graham et al. reference, p. 347. eq. 7.66. (Second edition, p. 361, eq. 7.67.) See also the Pólya-Szegő reference.
Also 4-Raney sequence. See the Graham et al. reference, pp. 346-347.
(End)
Bacher: "We describe the statistics of checkerboard triangulations obtained by coloring black every other triangle in triangulations of convex polygons." The current sequence (A002293) occurs on p. 12 as one of two "extremal sequences" of an array of coefficients of polynomials, whose generating functions are given in terms of hypergeometric functions. - Jonathan Vos Post, Oct 05 2007
A generating function in terms of a (labyrinthine) solution to a depressed quartic equation is given in the Copeland link for signed A005810. With D(z,t) that g.f., a g.f. for signed A002293 is {[-1+1/D(z,t)]/(4t)}^(1/3). - Tom Copeland, Oct 10 2012
For a relation to the inviscid Burgers's equation, see A001764. - Tom Copeland, Feb 15 2014
For relations to compositional inversion, the Legendre transform, and convex geometry, see the Copeland, the Schuetz and Whieldon, and the Gross (p. 58) links. - Tom Copeland, Feb 21 2017 (See also Gross et al. in A062994. - Tom Copeland, Dec 24 2019)
This is the number of A'Campo bicolored forests of degree n and co-dimension 0. This can be shown using generating functions or a combinatorial approach. See Combe and Jugé link below. - Noemie Combe, Feb 28 2017
Conjecturally, a(n) is the number of 3-uniform words over the alphabet [n] that avoid the patterns 231 and 221 (see the Defant and Kravitz link). - Colin Defant, Sep 26 2018
The compositional inverse o.g.f. pair in Copeland's comment above are related to a pair of quantum fields in Balduf's thesis by Theorem 4.2 on p. 92. Cf. A001764. - Tom Copeland, Dec 13 2019
a(n) is the total number of down steps before the first up step in all 3_1-Dyck paths of length 4*n. A 3_1-Dyck path is a lattice path with steps (1, 3), (1, -1) that starts and ends at y = 0 and stays above the line y = -1. - Sarah Selkirk, May 10 2020
a(n) is the number of pairs (A<=B) of noncrossing partitions of [2n] such that every block of A has exactly two elements. In fact, it is proved that a(n) is the number of planar tied arc diagrams with n arcs (see Aicardi link below). A planar diagram with n arcs represents a noncrossing partition A of [2n] with n blocks, each block containing the endpoints of one arc; each tie connects two arcs, so that the ties define a partition B >= A: the endpoints of two arcs connected by a tie belong to the same block of B. Ties do not cross arcs nor other ties iff B has a planar diagram, i.e., B is a noncrossing partition. - Francesca Aicardi, Nov 07 2022
Dropping the initial 1 (starting 1, 4, 22 with offset 1) yields the REVERT transformation 1, -4 ,10, -20, 35.. essentially A000292 without leading 0. - R. J. Mathar, Aug 17 2023
Number of rooted polyominoes composed of n pentagonal cells of the hyperbolic regular tiling with Schläfli symbol {5,oo}. A rooted polyomino has one external edge identified, and chiral pairs are counted as two. A stereographic projection of the {5,oo} tiling on the Poincaré disk can be obtained via the Christensson link. - Robert A. Russell, Jan 27 2024
This is instance k = 4 of the generalized Catalan family {C(k, n)}A130564.%20-%20_Wolfdieter%20Lang">{n>=0} given in a comment of A130564. - _Wolfdieter Lang, Feb 05 2024
a(n) is the cardinality of the planar ramified Jones monoid PR(J_n). - Diego Arcis, Nov 21 2024

Examples

			There are a(2) = 4 quartic trees (vertex degree <= 4 and 4 possible branchings) with 2 vertices (one of them the root). Adding one more branch (one more vertex) to these four trees yields 4*4 + 6 = 22 = a(3) such trees.

References

  • Miklos Bona, editor, Handbook of Enumerative Combinatorics, CRC Press, 2015, page 23.
  • R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 1990, pp. 200, 347.
  • Peter Hilton and Jean Pedersen, Catalan numbers, their generalization, and their uses, Math. Intelligencer 13 (1991), no. 2, 64-75.
  • V. A. Liskovets and T. R. Walsh, Enumeration of unrooted maps on the plane, Rapport technique, UQAM, No. 2005-01, Montreal, Canada, 2005.
  • G. Pólya and G. Szegő, Problems and Theorems in Analysis, Springer-Verlag, Heidelberg, New York, 2 vols., 1972, Vol. 1, problem 211, p. 146 with solution on p. 348.
  • 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).

Links

Crossrefs

Column k=3 of triangle A062993 and A070914.
Cf. A000260, A002295, A002296, A027836, A062994, A346646 (binomial transform), A346664 (inverse binomial transform).
Cf. A006632, A006633, A006634, A025174, A069271, A196678, A224274, A233658, A233666, A233667, A277877, A283049, A283101, A283102, A283103.
Polyominoes: A005038 (oriented), A005040 (unoriented), A369471 (chiral), A369472 (achiral), A001764 {4,oo}, A002294 {6,oo}.
Cf. A130564 (for generalized Catalan C(k, n), for = 4).

Programs

  • GAP
    List([0..22],n->Binomial(4*n,n)/(3*n+1)); # Muniru A Asiru, Nov 01 2018
  • Magma
    [ Binomial(4*n,n)/(3*n+1): n in [0..50]]; // Vincenzo Librandi, Apr 19 2011
  • Maple
    series(RootOf(g = 1+x*g^4, g),x=0,20); # Mark van Hoeij, Nov 10 2011
seq(binomial(4*n, n)/(3*n+1),n=0..20); # Robert FERREOL, Apr 02 2015
# Using the integral representation above:
Digits:=6;
R:=proc(x)((I + sqrt(3))*(4*sqrt(256 - 27*x) - 12*I*sqrt(3)*sqrt(x))^(1/3))/16 - ((I - sqrt(3))*(4*sqrt(256 - 27*x) + 12*I*sqrt(3)*sqrt(x))^(1/3))/16;end;
W:=proc(x) x^(-3/4) * sqrt(4*R(x) - 3^(3/4)*x^(1/4)/sqrt(R(x)))/(2*3^(1/4)*Pi);end;
# Attention: W(x) is singular at x = 0. Integration is done from  a very small positive x to x = 256/27.
# For a(8):  ... gives 420732
evalf(int(x^8*W(x),x=0.000001..256/27));
# Karol A. Penson, Jul 05 2024
  • Mathematica
    CoefficientList[InverseSeries[ Series[ y - y^4, {y, 0, 60}], x], x][[Range[2, 60, 3]]]
Table[Binomial[4n,n]/(3n+1),{n,0,25}] (* Harvey P. Dale, Apr 18 2011 *)
CoefficientList[1 + InverseSeries[Series[x/(1 + x)^4, {x, 0, 60}]], x] (* Gheorghe Coserea, Aug 12 2015 *)
terms = 22; A[] = 0; Do[A[x] = 1 + x*A[x]^4 + O[x]^terms, terms];
CoefficientList[A[x], x] (* Jean-François Alcover, Jan 13 2018 *)
  • PARI
    a(n)=binomial(4*n,n)/(3*n+1) /* Charles R Greathouse IV, Jun 16 2011 */
  • PARI
    my(x='x+O('x^33)); Vec(1 + serreverse(x/(1+x)^4)) \\ Gheorghe Coserea, Aug 12 2015
  • Python
    A002293_list, x = [1], 1
for n in range(100):
    x = x*4*(4*n+3)*(4*n+2)*(4*n+1)//((3*n+2)*(3*n+3)*(3*n+4))
    A002293_list.append(x) # Chai Wah Wu, Feb 19 2016

Formula

O.g.f. satisfies: A(x) = 1 + x*A(x)^4 = 1/(1 - x*A(x)^3).
a(n) = binomial(4*n,n-1)/n, n >= 1, a(0) = 1. From the Lagrange series of the o.g.f. A(x) with its above given implicit equation.
From Karol A. Penson, Apr 02 2010: (Start)
Integral representation as n-th Hausdorff power moment of a positive function on the interval [0, 256/27]:
a(n) = Integral_{x=0..256/27}(x^n((3/256) * sqrt(2) * sqrt(3) * ((2/27) * 3^(3/4) * 27^(1/4) * 256^(/4) * hypergeom([-1/12, 1/4, 7/12], [1/2, 3/4], (27/256)*x)/(sqrt(Pi) * x^(3/4)) - (2/27) * sqrt(2) * sqrt(27) * sqrt(256) * hypergeom([1/6, 1/2, 5/6], [3/4, 5/4], (27/256)*x)/ (sqrt(Pi) * sqrt(x)) - (1/81) * 3^(1/4) * 27^(3/4) * 256^(1/4) * hypergeom([5/12, 3/4, 13/12], [5/4, 3/2], (27/256)*x/(sqrt(Pi)*x^(1/4)))/sqrt(Pi))).
This representation is unique as it represents the solution of the Hausdorff moment problem.
O.g.f.: hypergeom([1/4, 1/2, 3/4], [2/3, 4/3], (256/27)*x);
E.g.f.: hypergeom([1/4, 1/2, 3/4], [2/3, 1, 4/3], (256/27)*x). (End)
a(n) = upper left term in M^n, M = the production matrix:
1, 1
3, 3, 1
6, 6, 3, 1
...
(where 1, 3, 6, 10, ...) is the triangular series. - Gary W. Adamson, Jul 08 2011
O.g.f. satisfies g = 1+x*g^4. If h is the series reversion of x*g, so h(x*g)=x, then (x-h(x))/x^2 is the o.g.f. of A006013. - Mark van Hoeij, Nov 10 2011
a(n) = binomial(4*n+1, n)/(4*n+1) = A062993(n+2,2). - Robert FERREOL, Apr 02 2015
a(n) = Sum_{i=0..n-1} Sum_{j=0..n-1-i} Sum_{k=0..n-1-i-j} a(i)*a(j)*a(k)*a(n-1-i-j-k) for n>=1; and a(0) = 1. - Robert FERREOL, Apr 02 2015
a(n) ~ 2^(8*n+1/2) / (sqrt(Pi) * n^(3/2) * 3^(3*n+3/2)). - Vaclav Kotesovec, Jun 03 2015
From Peter Bala, Oct 16 2015: (Start)
A(x)^2 is o.g.f. for A069271; A(x)^3 is o.g.f. for A006632;
A(x)^5 is o.g.f. for A196678; A(x)^6 is o.g.f. for A006633;
A(x)^7 is o.g.f. for A233658; A(x)^8 is o.g.f. for A233666;
A(x)^9 is o.g.f. for A006634; A(x)^10 is o.g.f. for A233667. (End)
D-finite with recurrence: a(n+1) = a(n)*4*(4*n + 3)*(4*n + 2)*(4*n + 1)/((3*n + 2)*(3*n + 3)*(3*n + 4)). - Chai Wah Wu, Feb 19 2016
E.g.f.: F([1/4, 1/2, 3/4], [2/3, 1, 4/3], 256*x/27), where F is the generalized hypergeometric function. - Stefano Spezia, Dec 27 2019
x*A'(x)/A(x) = (A(x) - 1)/(- 3*A(x) + 4) = x + 7*x^2 + 55*x^3 + 455*x^4 + ... is the o.g.f. of A224274. Cf. A001764 and A002294 - A002296. - Peter Bala, Feb 04 2022
a(n) = hypergeom([1 - n, -3*n], [2], 1). Row sums of A173020. - Peter Bala, Aug 31 2023
G.f.: t*exp(4*t*hypergeom([1, 1, 5/4, 3/2, 7/4], [4/3, 5/3, 2, 2], (256*t)/27))+1. - Karol A. Penson, Dec 20 2023
From Karol A. Penson, Jul 03 2024: (Start)
a(n) = Integral_{x=0..256/27} x^(n)*W(x)dx, n>=0, where W(x) = x^(-3/4) * sqrt(4*R(x) - 3^(3/4)*x^(1/4)/sqrt(R(x)))/(2*3^(1/4)*Pi), with R(x) = ((i + sqrt(3))*(4*sqrt(256 - 27*x) -12*i*sqrt(3*x))^(1/3))/16 - ((i - sqrt(3))*(4*sqrt(256 - 27*x) + 12*i* sqrt(3*x))^(1/3))/16, where i is the imaginary unit.
The elementary function W(x) is positive on the interval x = (0, 256/27) and is equal to the combination of hypergeometric functions in my formula from 2010; see above.
(Pi*W(x))^6 satisfies an algebraic equation of order 6, with integer polynomials as coefficients. (End)
G.f.: (Sum_{n >= 0} binomial(4*n+1, n)*x^n) / (Sum_{n >= 0} binomial(4*n, n)*x^n). - Peter Bala, Dec 14 2024
G.f. A(x) satisfies A(x) = 1/A(-x*A(x)^7). - Seiichi Manyama, Jun 16 2025

A000260 Number of rooted simplicial 3-polytopes with n+3 nodes; or rooted 3-connected triangulations with 2n+2 faces; or rooted 3-connected trivalent maps with 2n+2 vertices.

Original entry on oeis.org

1, 1, 3, 13, 68, 399, 2530, 16965, 118668, 857956, 6369883, 48336171, 373537388, 2931682810, 23317105140, 187606350645, 1524813969276, 12504654858828, 103367824774012, 860593023907540, 7211115497448720, 60776550501588855
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Comments

Number of rooted loopless planar maps with n edges. E.g., there are a(2)=3 loopless planar maps with 2 edges: two rooted paths (.-.-.) and one digon (.=.). - Valery A. Liskovets, Sep 25 2003
Number of intervals (i.e., ordered pairs (x,y) such that x<=y) in the Tamari lattice (rotation lattice of binary trees) of size n (see Pallo and Chapoton references). - Ralf Stephan, May 08 2007, Jean Pallo (Jean.Pallo(AT)u-bourgogne.fr), Sep 11 2007
Number of rooted triangulations of type [n, 0] (see Brown paper eq (4.8)). - Michel Marcus, Jun 23 2013
Equivalently, number of rooted bridgeless planar maps with n edges. - Noam Zeilberger, Oct 06 2016
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
Number of uniquely sorted permutations of [2n+1] that avoid the pattern 231. Also the number of uniquely sorted permutations of [2n+1] that avoid 132. - Colin Defant, Jun 13 2019
The sequence 1,3,13,68,... appears naturally in integral geometry, namely in the algebra of unitarily invariant valuations on complex space forms. - Andreas Bernig, Feb 02 2020

Examples

			G.f. = 1 + x + 3*x^2 + 13*x^3 + 68*x^4 + 399*x^5 + 2530*x^6 + 16965*x^7 + ...

References

  • C. F. Earl and L. J. March, Architectural applications of graph theory, pp. 327-355 of R. J. Wilson and L. W. Beineke, editors, Applications of Graph Theory. Academic Press, NY, 1979.
  • J. L. Gross and J. Yellen, eds., Handbook of Graph Theory, CRC Press, 2004; p. 714.
  • Handbook of Combinatorics, North-Holland '95, p. 891.
  • 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).
  • W. T. Tutte, The enumerative theory of planar maps, in A Survey of Combinatorial Theory (J. N. Srivastava et al. eds.), pp. 437-448, North-Holland, Amsterdam, 1973.

Links

Crossrefs

Row sums of A342981.
Column 0 of A146305 and A341856; Column 2 of A255918.
Sequences mentioned in the Noam Zeilberger 2018 video: A000168, A000260, A000309, A000698, A000699, A002005, A062980, A267827.
Cf. A000256, A006013, A027836, A069271, A263191, A290326.

Programs

  • Magma
    [Binomial(4*n+1, n+1)-9*Binomial(4*n+1, n-1): n in [0..25]]; // Vincenzo Librandi, Nov 24 2016
  • Maple
    A000260 := proc(n)
    2*(4*n+1)!/((n+1)!*(3*n+2)!) ;
end proc:
  • Mathematica
    Table[Binomial[4n+1,n+1]-9*Binomial[4n+1,n-1],{n,0,25}] (* Harvey P. Dale, Aug 23 2011 *)
a[ n_] := SeriesCoefficient[ HypergeometricPFQ[ {1/2, 3/4, 1, 5/4}, {4/3, 5/3, 2}, 256/27 x], {x, 0, n}]; (* Michael Somos, Dec 23 2014 *)
terms = 22; G[] = 0; Do[G[x] = 1+x*G[x]^4 + O[x]^terms, terms];
CoefficientList[(2-G[x])*G[x]^2, x] (* Jean-François Alcover, Jan 13 2018, after Mark van Hoeij *)
  • PARI
    {a(n) = if( n<0, 0, 2 * (4*n + 1)! / ((n + 1)! * (3*n + 2)!))}; /* Michael Somos, Sep 07 2005 */
  • PARI
    {a(n) = binomial( 4*n + 2, n + 1) / ((2*n + 1) * (3*n + 2))}; /* Michael Somos, Mar 28 2012 */
  • Sage
    def a(n):
    n = ZZ(n)
    return (4*n + 2).binomial(n + 1) // ((2*n + 1) * (3*n + 2))
# F. Chapoton, Aug 06 2015

Formula

a(n) = 2*(4*n+1)! / ((n+1)!*(3*n+2)!) = binomial(4*n+1, n+1) - 9*binomial(4*n+1, n-1).
G.f.: (2-g)*g^2 where g = 1 + x*g^4 is the g.f. of A002293. - Mark van Hoeij, Nov 10 2011
G.f.: hypergeom([1,1/2,3/4,5/4],[2,4/3,5/3],256*x/27) = 1 + 120*x/(Q(0)-120*x); Q(k) = 8*x*(2*k+1)*(4*k+3)*(4*k+5) + 3*(k+2)*(3*k+4)*(3*k+5) - 24*x*(k+2)*(2*k+3)*(3*k+4)*(3*k+5)*(4*k+7)*(4*k+9)/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, Nov 25 2011
a(n) = binomial(4*n + 2, n + 1) / ((2*n + 1) * (3*n + 2)). - Michael Somos, Mar 28 2012
a(n) * (n+1) = A069271(n). - Michael Somos, Mar 28 2012
0 = F(a(n), a(n+1), ..., a(n+8)) for all n in Z where a(-1) = 3/4 and F() is a polynomial of degree 2 with integer coefficients and 29 monomials. - Michael Somos, Dec 23 2014
D-finite with recurrence: 3*(3*n+2)*(3*n+1)*(n+1)*a(n) - 8*(4*n+1)*(2*n-1)*(4*n-1)*a(n-1) = 0. - R. J. Mathar, Oct 21 2015
a(n) = Sum_{k=1..A000108(n)} k * A263191(n,k). - Alois P. Heinz, Nov 16 2015
a(n) ~ 2^(8*n+7/2) / (sqrt(Pi) * n^(5/2) * 3^(3*n+5/2)). - Vaclav Kotesovec, Feb 26 2016
E.g.f.: 3F3(1/2,3/4,5/4; 4/3,5/3,2; 256*x/27). - Ilya Gutkovskiy, Feb 01 2017
From Gheorghe Coserea, Aug 17 2017: (Start)
G.f. y(x) satisfies:
0 = x^3*y^4 + 3*x^2*y^3 + x*(8*x+3)*y^2 - (20*x-1)*y + 16*x-1.
0 = x*(256*x - 27)*deriv(y,x) - 8*x^2*y^3 - 25*x*y^2 + 4*(24*x-11)*y + 44.
(End)
From Karol A. Penson, Apr 06 2022: (Start)
a(n) = Integral_{x=0...256/27} x^n*W(x), where
W(x) = (sqrt(2)/Pi)*(h1(x) - h2(x) + h3(x)) and
h1(x) = 3F2([-6/12,-2/12, 2/12], [ 3/12, 9/12], (27*x)/256)/((x/2)^(1/2));
h2(x) = 3F2([-3/12, 1/12, 5/12], [ 6/12, 15/12], (27*x)/256)/(x^(1/4));
h3(x) = 3F2([ 3/12, 7/12, 11/12], [18/12, 21/12], (27*x)/256)/(x^(-1/4)*32).
This integral representation is unique as the solution of n-th Hausdorff power moment of the function W. Using only the definition of a(n), W(x) can be proven to be positive. W(x) is singular at x = 0 and for x > 0 is monotonically decreasing to zero at x = 256/27. (End)
a(n) = (1/27^n) * Product_{1 <= i <= j <= 3*n} (3*i + j + 3)/(3*i + j - 1). Cf. A006013. - Peter Bala, Feb 21 2023

Extensions

Edited by F. Chapoton, Feb 03 2011

A342981 Triangle read by rows: T(n,k) is the number of rooted planar maps with n edges, k faces and no isthmuses, n >= 0, k = 1..n+1.

Original entry on oeis.org

1, 0, 1, 0, 1, 2, 0, 1, 7, 5, 0, 1, 16, 37, 14, 0, 1, 30, 150, 176, 42, 0, 1, 50, 449, 1104, 794, 132, 0, 1, 77, 1113, 4795, 7077, 3473, 429, 0, 1, 112, 2422, 16456, 41850, 41504, 14893, 1430, 0, 1, 156, 4788, 47832, 189183, 319320, 228810, 63004, 4862
Offset: 0

Views

Author

Andrew Howroyd, Apr 02 2021

Keywords

Comments

The number of vertices is n + 2 - k.
For k >= 2, column k is a polynomial of degree 3*(k-2). This is because adding a face can increase the number of vertices whose degree is greater than two by at most two.
By duality, also the number of loopless rooted planar maps with n edges and k vertices.

Examples

			Triangle begins:
  1;
  0, 1;
  0, 1,   2;
  0, 1,   7,    5;
  0, 1,  16,   37,    14;
  0, 1,  30,  150,   176,    42;
  0, 1,  50,  449,  1104,   794,   132;
  0, 1,  77, 1113,  4795,  7077,  3473,   429;
  0, 1, 112, 2422, 16456, 41850, 41504, 14893, 1430;
  ...

Links

Crossrefs

Columns k=3..4 are A005581, A006468.
Diagonals are A000108, A006419, A006420, A006421.
Row sums are A000260.
Cf. A002293, A027836, A082680, A269920, A342980, A343092.

Programs

  • Mathematica
    G[m_, y_] := Sum[x^n*Sum[(n + k - 1)!*(2*n - k)!*y^k/(k!*(n + 1 - k)!*(2*k - 1)!*(2*n - 2*k + 1)!), {k, 1, n}], {n, 1, m}] + O[x]^m;
H[n_] := With[{g = 1 + x*y + x*G[n - 1, y]}, Sqrt[InverseSeries[x/g^2 + O[x]^(n + 1), x]/x]];
CoefficientList[#, y]& /@ CoefficientList[H[10], x] // Flatten (* Jean-François Alcover, Apr 15 2021, after Andrew Howroyd *)
  • PARI
    \\ here G(n, y) gives A082680 as g.f.
G(n,y)={sum(n=1, n, x^n*sum(k=1, n, (n+k-1)!*(2*n-k)!*y^k/(k!*(n+1-k)!*(2*k-1)!*(2*n-2*k+1)!))) + O(x*x^n)}
H(n)={my(g=1+x*y+x*G(n-1, y), v=Vec(sqrt(serreverse(x/g^2)/x))); vector(#v, n, Vecrev(v[n], n))}
{ my(T=H(8)); for(n=1, #T, print(T[n])) }

Formula

G.f. A(x,y) satisfies A(x) = G(x*A(x,y)^2, y) where G(x,y) = 1 + x*y + x*B(x,y) and B(x,y) is the g.f. of A082680.
A027836(n+1) = Sum_{k=1..n+1} k*T(n,k).
A002293(n) = Sum_{k=1..n+1} k*T(n,n+2-k).

A103942 Number of unrooted n-edge isthmusless maps in the plane (planar with a distinguished outside face).

Original entry on oeis.org

1, 1, 3, 9, 38, 187, 1120, 7083, 47990, 337676, 2455517, 18310155, 139447034, 1080773098, 8502896424, 67763884363, 546147639926, 4445389286380, 36501274080076, 302060508150976, 2517213486505592, 21110062391001119, 178052027949519768, 1509631210682469661, 12860805940582898474
Offset: 0

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Author

Valery A. Liskovets, Mar 17 2005

Keywords

References

  • V. A. Liskovets and T. R. Walsh, Enumeration of unrooted maps on the plane, Rapport technique, UQAM, No. 2005-01, Montreal, Canada, 2005.

Links

Crossrefs

Cf. A027836, A006390, A103941, A000260.

Programs

  • Mathematica
    a[n_] := (1/(2n)) ((5n^2 + 13n + 2) Binomial[4n, n]/((n+1)(3n+1)(3n+2)) + Sum[Boole[0 < k < n] EulerPhi[n/k] Binomial[4k, k], {k, Divisors[n]}] + q[n]);
q[n_] := If[EvenQ[n], 0, (n-1) Binomial[2n, (n-1)/2]]/(n+1);
Array[a, 20] (* Jean-François Alcover, Sep 01 2019 *)
  • PARI
    a(n) = {if(n==0, 1, (sumdiv(n, d, if(dAndrew Howroyd, Mar 28 2021

Formula

For n > 0, a(n) = (1/(2n))*[(5n^2+13n+2)*binomial(4n, n)/((n+1)(3n+1)(3n+2)) + Sum_{0A000010), q(n)=0 if n is even and q(n)=(n-1)*binomial(2n, (n-1)/2)/(n+1) if n is odd.

Extensions

a(0)=1 prepended and terms a(21) and beyond from Andrew Howroyd, Mar 28 2021
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