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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). 10
 1, 5, 60, 1105, 27120, 828250, 30220800, 1282031525, 61999046400, 3366961243750, 202903221120000, 13437880555850250, 970217083619328000, 75849500508999712500, 6383483988812390400000, 575440151532675686278125 (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 [Zeilberger]. - Pierre Lescanne, Feb 26 2017 Proved (bijection) by O. Bodini, D. Gardy, A. Jacquot (2013). - Olivier Bodini, Mar 30 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] 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] 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)] 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. 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 and Alain Giorgetti, A correspondence between rooted planar maps and normal planar lambda terms, 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 xrange(1, n - 1)]) print map(a, xrange(21)) # Indranil Ghosh, Aug 09 2017 CROSSREFS 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 Alexandre Vidal (samuel.vidal(AT)free.fr), Mar 30 2007 STATUS approved

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Last modified May 24 20:44 EDT 2018. Contains 304537 sequences. (Running on oeis4.)