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A062980 a(0) = 1, a(1) = 5; for n > 1, a(n) = 6*n*a(n-1) + Sum_{k=1..n-2} a(k)*a(n-k-1). 18
1, 5, 60, 1105, 27120, 828250, 30220800, 1282031525, 61999046400, 3366961243750, 202903221120000, 13437880555850250, 970217083619328000, 75849500508999712500, 6383483988812390400000, 575440151532675686278125, 55318762960656722780160000 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

Number of rooted unlabeled connected triangular maps on a compact closed oriented surface with 2n faces (and thus 3n edges). [Vidal]

Equivalently, the number of pair of permutations (sigma,tau) up to simultaneous conjugacy on a pointed set of size 6*n with sigma^3=tau^2=1, acting transitively and with no fixed point. [Vidal]

Also, the asymptotic expansion of the Airy function Ai'(x)/Ai(x) = -sqrt(x) - 1/(4x) + Sum_{n>=2} (-1)^n a(n) (4x)^ (1/2-3n/2). [Praehofer]

Maple 6 gives the wrong asymptotics of Ai'(x)=AiryAi(1,x) as x->infty apart from the 3rd term. Therefore asympt(AiryAi(1,x/4)/AiryAi(x/4),x); reproduces only the value a(1)=1 correctly.

Number of closed linear lambda terms (see [Bodini, Gardy, Jacquot, 2013] and [N. Zeilberger, 2015] references). - Pierre Lescanne, Feb 26 2017

Proved (bijection) by O. Bodini, D. Gardy, A. Jacquot (2013). - Olivier Bodini, Mar 30 2018

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

LINKS

Reinhard Zumkeller, Table of n, a(n) for n = 0..100

O. Bodini, D. Gardy, A. Jacquot, Asymptotics and random sampling for BCI and BCK lambda terms, Theor. Comput. Sci. 502: 227-238 (2013).

Jørgen Ellegaard Andersen, Gaëtan Borot, Leonid O. Chekhov and Nicolas Orantin, The ABCD of topological recursion, arXiv:1703.03307 [math-ph], 2017. [See Appendix A.1]

Laura Ciobanu and Alexander Kolpakov, Free subgroups of free products and combinatorial hypermaps, arXiv:1708.03842 [math.CO], 2017.

P. Cvitanovic, B. Lautrup and R. B. Pearson, The number and weights of Feynman diagrams, Phys. Rev. D18, (1978), 1939-1949, (eq 3.14 and fig 1b).

Bertrand Eynard, Topological expansion for the 1-hermitian matrix model correlation functions, Journal of High Energy Physics 11 (2004). [See Eq. (5.12) and Appendix A]

S. R. Finch, Shapes of binary trees, June 24, 2004. [Cached copy, with permission of the author]

Éric Fusy, Luca Lionni, Adrian Tanasa, Combinatorial study of graphs arising from the Sachdev-Ye-Kitaev model, arXiv:1810.02146 [math.CO], 2018.

Katarzyna Grygiel, Pawel M. Idziak and Marek Zaionc, How big is BCI fragment of BCK logic, arXiv preprint arXiv:1112.0643 [cs.LO], 2011. [From N. J. A. Sloane, Feb 21 2012]

George Kaye, A visualiser for linear lambda-terms as 3-valent rooted maps, University of Birmingham (UK, 2019).

S. Janson, The Wiener index of simply generated random trees, Random Structures Algorithms 22 (2003) 337-358.

Michael J. Kearney, Satya N. Majumdar and Richard J. Martin, The first-passage area for drifted Brownian motion and the moments of the Airy distribution, arXiv:0706.2038 [cond-mat.stat-mech], 2007. [a(n) = 8^n * K_n from Eq. (3)]

Pierre Lescanne, Quantitative aspects of linear and affine closed lambda terms, arXiv:1702.03085 [cs.DM], 2017. Also in ACM Transactions on Computational Logic (TOCL 2019), Vol. 19, No. 2, Article No. 9.

R. J. Martin and M. J. Kearney, An exactly solvable self-convolutive recurrence, Aequat. Math., 80 (2010), 291-318. See p. 292.

NIST Digital Library of Mathematical Functions, Airy Functions.

A. N. Stokes, Continued fraction solutions of the Riccati equation, Bull. Austral. Math. Soc. Vol. 25 (1982), 207-214.

Paul Tarau, Valeria de Paiva, Deriving Theorems in Implicational Linear Logic, Declaratively, arXiv:2009.10241 [cs.LO], 2020. See also Github, (2020).

Samuel Vidal and Michel Petitot, Counting Rooted and Unrooted Triangular Maps, Journal of Nonlinear Systems and Applications (2009) 151-154.

Eric Weisstein's World of Mathematics, Airy Functions, contains the definitions of Ai(x), Bi(x).

Noam Zeilberger, Counting isomorphism classes of beta-normal linear lambda terms, arXiv:1509.07596 [cs.LO], 2015.

Noam Zeilberger, Linear lambda terms as invariants of rooted trivalent maps, arXiv preprint 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 [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.

Noam Zeilberger and Alain Giorgetti, A correspondence between rooted planar maps and normal planar lambda terms, arXiv:1408.5028 [cs.LO], 2014-2015; Logical Methods in Computer Science, vol. 11 (3:22), 2015, pp. 1-39.

FORMULA

With offset 1, then a(1) = 1 and, for n > 1, a(n) = (6*n-8)*a(n-1) + Sum_{k=1..n-1} a(k)*a(n-k) [Praehofer] [Martin and Kearney].

a(n) = 6/Pi^2*int((4*x)^(3*n/2)/(Ai(x)^2+Bi(x)^2), x=0..inf). - Vladimir Reshetnikov, Sep 24 2013

a(n) ~ 3 * 6^n * n! / Pi. - Vaclav Kotesovec, Jan 19 2015

0 = 6*x^2*y' + x*y^2 + (4*x-1)*y + 1, where y(x) = Sum_{n>=0} a(n)*x^n. - Gheorghe Coserea, Apr 02 2017

From Peter Bala, May 21 2017: (Start)

G.f. as an S-fraction: A(x) = 1/(1 - 5*x/(1 - 7*x/(1 - 11*x/(1 - 13*x/(1 - ... - (6*n - 1)*x/(1 - (6*n + 1)*x/(1 - .... See Stokes.

x*A(x) = B(x)/(1 + 2*B(x)), where B(x) = x + 7*x^2 + 84*x^3 + 1463*x^4 + ... is the o.g.f. of A172455.

A(x) = 1/(1 + 2*x - 7*x/(1 - 5*x/(1 - 13*x/(1 - 11*x/(1 - ... - (6*n + 1)*x/(1 - (6*n - 1)*x/(1 - .... (End)

EXAMPLE

1 + 5*x + 60*x^2 + 1105*x^3 + 27120*x^4 + 828250*x^5 + 30220800*x^6 + ...

MAPLE

a:= proc(n) option remember; `if`(n<2, 4*n+1,

      6*n*a(n-1) +add(a(k)*a(n-k-1), k=1..n-2))

    end:

seq(a(n), n=0..25);  # Alois P. Heinz, Mar 31 2017

MATHEMATICA

max = 16; f[y_] := -Sqrt[x] - 1/(4*x) + Sum[(-1)^n*a[n]*(4*x)^(1/2 - 3*(n/2)), {n, 2, max}] /. x -> 1/y^2; s[y_] := Normal[ Series[ AiryAiPrime[x] / AiryAi[x], {x, Infinity, max + 7}]] /. x -> 1/y^2; sol = SolveAlways[ Simplify[ f[y] == s[y], y > 0], y] // First; Join[{1, 5}, Table[a[n], {n, 3, max}] /. sol] (* Jean-François Alcover, Oct 09 2012, from Airy function asymptotics *)

a[0] = 1; a[n_] := a[n] = (6*(n-1)+4)*a[n-1] + Sum[a[i]*a[n-i-1], {i, 0, n-1}]; Table[a[n], {n, 0, 15}] (* Jean-François Alcover, Nov 29 2013, after Vladimir Reshetnikov *)

PROG

(PARI) {a(n) = local(A); n++; if( n<1, 0, A = vector(n); A[1] = 1; for( k=2, n, A[k] = (6*k - 8) * A[k-1] + sum( j=1, k-1, A[j] * A[k-j])); A[n])} /* Michael Somos, Jul 24 2011 */

(Haskell)

a062980 n = a062980_list !! n

a062980_list = 1 : 5 : f 2 [5, 1] where

   f u vs'@(v:vs) = w : f (u + 1) (w : vs') where

     w = 6 * u * v + sum (zipWith (*) vs_ $ reverse vs_)

     vs_ = init vs

-- Reinhard Zumkeller, Jun 03 2013

(Python)

from sympy.core.cache import cacheit

@cacheit

def a(n): return n*4 + 1 if n<2 else 6*n*a(n - 1) + sum(a(k)*a(n - k - 1) for k in range(1, n - 1))

print([a(n) for n in range(21)]) # Indranil Ghosh, Aug 09 2017

CROSSREFS

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

Pointed version of A012114. Connected pointed version of A012115. [Xrefs A012114 and A012115 are probably wrong. - Vaclav Kotesovec, Jan 27 2015]

Cf. A060506, A060507, A094199, A121350, A121352, A005133, A172455.

Sequence in context: A260776 A128574 A120976 * A113665 A147585 A138215

Adjacent sequences:  A062977 A062978 A062979 * A062981 A062982 A062983

KEYWORD

nonn,nice,easy

AUTHOR

Michael Praehofer (praehofer(AT)ma.tum.de), Jul 24 2001

EXTENSIONS

Entry revised by N. J. A. Sloane based on comments from Samuel A. Vidal, Mar 30 2007

STATUS

approved

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Last modified April 10 23:13 EDT 2021. Contains 342877 sequences. (Running on oeis4.)