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).
1, 5, 60, 1105, 27120, 828250, 30220800, 1282031525, 61999046400, 3366961243750, 202903221120000, 13437880555850250, 970217083619328000, 75849500508999712500, 6383483988812390400000, 575440151532675686278125, 55318762960656722780160000
Offset: 0
Examples
1 + 5*x + 60*x^2 + 1105*x^3 + 27120*x^4 + 828250*x^5 + 30220800*x^6 + ...
Links
- Reinhard Zumkeller, Table of n, a(n) for n = 0..100
- O. Bodini, D. Gardy, and 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). DOI:10.1103/PhysRevD.18.1939
- 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]
- Steven R. Finch, Shapes of binary trees, June 24, 2004. [Cached copy, with permission of the author]
- Steven R. Finch, An exceptional convolutional recurrence, arXiv:2408.12440 [math.CO], 2024.
- 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]
- Kristian Gustavsson and Bernhard Mehlig, Statistical models for spatial patterns of heavy particles in turbulence, arxiv:1412.4374 [physics.flu-dyn], 2014-2016. [See Figure 14]
- M. J. Kearny and R. J. Martin, Normalized functionals of first passage Brownian motion and a curious connection with the maximal relative height of fluctuating interfaces, J. Phys. A, Math. Theor. 49, No. 19, Article ID 195001, 20 p. (2016).
- 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 and 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.
Crossrefs
Programs
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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 -
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 *) -
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 */ -
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
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)*Integral_{x=0..oo} ((4*x)^(3*n/2)/(Ai(x)^2 + Bi(x)^2)) dt. - 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)
Extensions
Entry revised by N. J. A. Sloane based on comments from Samuel A. Vidal, Mar 30 2007
Comments