A266240 -id:A266240 - cp's OEIS frontend

A269476 Column 4 of A266240.

56581525, 11100235520, 1191262520280, 92809670660096, 5875149131469024, 320744781170737152, 15658287081481903872, 700135726021459574784, 29159104554804741742080, 1145151298823440950099968, 42804391324225851826851840, 1533861483718086352674226176

Offset: 7

Vaclav Kotesovec, Feb 27 2016

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 7..740

Cf. A266240.

Recurrence: (n-8)*(n-7)*(n-6)*(8865313032240*n^7 - 317083364299800*n^6 + 4684927154541525*n^5 - 36954847894577400*n^4 + 167489792744727732*n^3 - 434509931893116160*n^2 + 594863631325194528*n - 329506352166682880)*a(n) = 48*(n-8)*(159575634580320*n^9 - 5627712740106240*n^8 + 82146941258133330*n^7 - 643013667125582535*n^6 + 2917921647785653476*n^5 - 7719296227286609376*n^4 + 11246681386937347392*n^3 - 7567478663869195664*n^2 + 942570862638469952*n + 715530999520296960)*a(n-2) - 20736*(n-2)*(3*n - 10)*(3*n - 8)*(8865313032240*n^7 - 192968981848440*n^6 + 1624613077652085*n^5 - 6648290558122950*n^4 + 13479332779855932*n^3 - 11835869605572088*n^2 + 2032110719189792*n + 1581052254093312)*a(n-4).

a(n) ~ 37079 * 2^(2*n - 21/2) * 3^(3*n/2 - 11/2) * n^(15/2) / (3575*sqrt(Pi)).

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

A269418 a(n) is numerator of y(n), where y(n+1) = (25n^2-1)/48 y(n) + (1/2)Sum_{k=1..n}y(k)y(n+1-k), with y(0) = -1.

Original entry on oeis.org

-1, 1, 49, 1225, 4412401, 73560025, 245229441961, 7759635184525, 2163099334469560445, 243352176577765537625, 126154825844683612669806743, 307996788703417873806157775, 3816216508144039222348410175181221, 4472139245793702477426700875742975

Offset: 0

Gheorghe Coserea, Feb 25 2016

			For n=0 we have t(0) = (-1) / (2^(-2)*gamma(-1/2)) = 2/sqrt(Pi).
For n=1 we have t(1) = (1/48) / (2^(-1)*gamma(2))  = 1/24.
n   y(n)                        t(n)
0   -1                          2/sqrt(Pi)
1   1/48                        1/24
2   49/4608                     7/(4320*sqrt(Pi))
3   1225/55296                  245/15925248
4   4412401/42467328            37079/(96074035200*sqrt(Pi))
5   73560025/84934656           38213/14089640214528
6   245229441961/21743271936    5004682489/(92499927372103680000*sqrt(Pi))
7   7759635184525/36691771392   6334396069/20054053184087387013120
...

Gheorghe Coserea, Table of n, a(n) for n = 0..187
Edward A. Bender, Zhicheng Gao, L. Bruce Richmond, The map asymptotics constant tg, The Electronic Journal of Combinatorics, Volume 15 (2008), Research Paper #R51.
Stavros Garoufalidis, Thang T.Q. Le, Marcos Marino, Analyticity of the Free Energy of a Closed 3-Manifold, arXiv:0809.2572 [math.GT], 2008.

Cf. A266240, A269419 (denominator).

y[0] = -1;
y[n_] := y[n] = (25(n-1)^2-1)/48 y[n-1] + 1/2 Sum[y[k] y[n-k], {k, 1, n-1}];
Table[y[n] // Numerator, {n, 0, 13}] (* Jean-François Alcover, Oct 23 2018 *)

seq(n) = {
  my(y  = vector(n));
  y[1] = 1/48;
  for (g = 1, n-1,
       y[g+1] = (25*g^2-1)/48 * y[g] + 1/2*sum(k = 1, g, y[k]*y[g+1-k]));
  return(concat(-1,y));
}
apply(numerator, seq(13))

t(g) = (A269418(g)/A269419(g)) / (2^(g-2) * gamma((5*g-1)/2)), where t(g) is the orientable map asymptotics constant and gamma is the Gamma function.

A269419 a(n) is denominator of y(n), where y(n+1) = (25n^2-1)/48 y(n) + (1/2)Sum_{k=1..n}y(k)y(n+1-k), with y(0) = -1.

Original entry on oeis.org

1, 48, 4608, 55296, 42467328, 84934656, 21743271936, 36691771392, 400771988324352, 1352605460594688, 16620815899787526144, 779100745302540288, 153177439332441840943104, 2393397489569403764736, 235280546814630667688607744, 57441539749665690353664

Offset: 0

Gheorghe Coserea, Feb 25 2016

			For n=0 we have t(0) = (-1) / (2^(-2)*gamma(-1/2)) = 2/sqrt(Pi).
For n=1 we have t(1) = (1/48) / (2^(-1)*gamma(2))  = 1/24.
n   y(n)                        t(n)
0   -1                          2/sqrt(Pi)
1   1/48                        1/24
2   49/4608                     7/(4320*sqrt(Pi))
3   1225/55296                  245/15925248
4   4412401/42467328            37079/(96074035200*sqrt(Pi))
5   73560025/84934656           38213/14089640214528
6   245229441961/21743271936    5004682489/(92499927372103680000*sqrt(Pi))
7   7759635184525/36691771392   6334396069/20054053184087387013120
...

Gheorghe Coserea, Table of n, a(n) for n = 0..500
Edward A. Bender, Zhicheng Gao, L. Bruce Richmond, The map asymptotics constant tg, The Electronic Journal of Combinatorics, Volume 15 (2008), Research Paper #R51.
Stavros Garoufalidis, Thang T.Q. Le, Marcos Marino, Analyticity of the Free Energy of a Closed 3-Manifold, arXiv:0809.2572 [math.GT], 2008.

Cf. A266240, A269418 (numerator).

y[0] = -1;
y[n_] := y[n] = (25(n-1)^2-1)/48 y[n-1] + 1/2 Sum[y[k] y[n-k], {k, 1, n-1}];
Table[y[n] // Denominator, {n, 0, 15}] (* Jean-François Alcover, Oct 23 2018 *)

seq(n) = {
  my(y  = vector(n));
  y[1] = 1/48;
  for (g = 1, n-1,
       y[g+1] = (25*g^2-1)/48 * y[g] + 1/2*sum(k = 1, g, y[k]*y[g+1-k]));
  return(concat(-1,y));
}
apply(denominator, seq(14))

t(g) = (A269418(g)/A269419(g)) / (2^(g-2) * gamma((5*g-1)/2)), where t(g) is the orientable map asymptotics constant and gamma is the Gamma function.

A269473 a(n) is the number of rooted 2n-face triangulations in an orientable surface of genus 1.

Original entry on oeis.org

1, 28, 664, 14912, 326496, 7048192, 150820608, 3208396800, 67968706048, 1435486650368, 30246600953856, 636154755940352, 13360333295173632, 280258138588839936, 5873204471357374464, 122980760637407232000, 2573349967992101142528, 53815038103588370907136

Offset: 1

Vaclav Kotesovec, Feb 27 2016

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 1..750
I. P. Goulden and D. M. Jackson, The KP hierarchy, branched covers, and triangulations, Advances in Mathematics, Volume 219, Issue 3, 20 October 2008, Pages 932-951.
Evgeniy Krasko, Alexander Omelchenko, Enumeration of r-regular Maps on the Torus. Part I: Enumeration of Rooted and Sensed Maps, arXiv:1709.03225 [math.CO], 2017.
Evgeniy Krasko, 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, Volume 342, Issue 2, February 2019, Pages 584-599.

Column k=1 of A266240.

Rest[CoefficientList[InverseSeries[Series[Sqrt[1728 - 432/x + (30*Sqrt[1 + 48*x] - 54)/x^2 + (Sqrt[1 + 48*x] - 1)/x^3]/864, {x, 0, 20}], x], x]] (* Vaclav Kotesovec, Jul 28 2018 *)

factorial2(n) = my(x = (2^(n\2)*(n\2)!)); if (n%2, n!/x, x);
a(n) = {
  my(f2 = factorial2);
  4^(n-1)*f2(n-1)/n! * sum(k=0, n-1, 3^k * f2(3*n-2*k-2)/(n-1-k)!);
};
\\ test: y='x*Ser(vector(303, n, a(n))); 0 == 2*(432*x^2 - 1)^2*y^3 + (432*x^2 - 1)*y^2 + 54*x^2*y + x^2
\\ Gheorghe Coserea, Jul 27 2018

Recurrence: (n-1)*n*(15*n - 46)*a(n) = 48*(270*n^3 - 1503*n^2 + 2478*n - 1280)*a(n-2) - 20736*(3*n - 10)*(3*n - 8)*(15*n - 16)*a(n-4).
a(n) ~ 2^(2*n-3) * 3^(3*n/2).
From Gheorghe Coserea, Jul 27 2018: (Start)
a(n+1) = 4^n * n!!/(n+1)! * Sum_{k=0..n} 3^k*(3*n-2*k+1)!!/(n-k)!. (see Krasko link)
G.f. y(x) satisfies:
0 = 2*(432*x^2 - 1)^2*y^3 + (432*x^2 - 1)*y^2 + 54*x^2*y + x^2.
0 = x*(432*x^2 - 1)*(108*x^2 + 1)*deriv(y,x) + 2*(432*x^2 - 1)*(648*x^2 + 1)*y^2 + (31104*x^4 + 1116*x^2 + 1)*y + 30*x^2.
0 = (5184*x^2 - 7)*(432*x^2 - 1)^2*y''' + 1296*x*(432*x^2 - 1)*(12096*x^2 - 13)*y'' + 48*(199314432*x^4 - 479088*x^2 + 581)*y' + 663552*x*(2592*x^2 - 11)*y.
(End)

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A269474 Column 2 of A266240.

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A269475 Column 3 of A266240.

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A269476 Column 4 of A266240.

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

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A269418 a(n) is numerator of y(n), where y(n+1) = (25*n^2-1)/48 * y(n) + (1/2)*Sum_{k=1..n}y(k)*y(n+1-k), with y(0) = -1.

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A269419 a(n) is denominator of y(n), where y(n+1) = (25*n^2-1)/48 * y(n) + (1/2)*Sum_{k=1..n}y(k)*y(n+1-k), with y(0) = -1.

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A269473 a(n) is the number of rooted 2n-face triangulations in an orientable surface of genus 1.

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A269418 a(n) is numerator of y(n), where y(n+1) = (25n^2-1)/48 y(n) + (1/2)Sum_{k=1..n}y(k)y(n+1-k), with y(0) = -1.

A269419 a(n) is denominator of y(n), where y(n+1) = (25n^2-1)/48 y(n) + (1/2)Sum_{k=1..n}y(k)y(n+1-k), with y(0) = -1.