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A267827 Number of closed indecomposable linear lambda terms with 2n+1 applications and abstractions.

%I A267827 #64 Jun 19 2020 14:33:36
%S A267827 1,2,20,352,8624,266784,9896448,426577920,20918138624,1149216540160,
%T A267827 69911382901760,4665553152081920,338942971881472000,
%U A267827 26631920159494995968,2250690001888540950528,203595258621775065120768,19629810220331494121865216
%N A267827 Number of closed indecomposable linear lambda terms with 2n+1 applications and abstractions.
%C A267827 A linear lambda term is indecomposable if it has no closed proper subterm.
%C A267827 Equivalently, number of closed bridgeless rooted trivalent maps (on compact oriented surfaces of arbitrary genus) with 2n+1 trivalent vertices (and 1 univalent vertex).
%C A267827 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
%H A267827 Gheorghe Coserea, <a href="/A267827/b267827.txt">Table of n, a(n) for n = 0..303</a>
%H A267827 Lawrence Dresner, <a href="http://www.osti.gov/scitech/biblio/6829819">Protection of a test magnet wound with a Ag/BSCCO high-temperature superconductor</a>, Oak Ridge National Lab technical report (ORNL/HTSPC-3), 1992. See Eq. (25).
%H A267827 Noam Zeilberger, <a href="http://arxiv.org/abs/1512.06751">Linear lambda terms as invariants of rooted trivalent maps</a>, arXiv:1512.06751 [cs.LO], 2015.
%H A267827 Noam Zeilberger, <a href="https://arxiv.org/abs/1804.10540">A theory of linear typings as flows on 3-valent graphs</a>, arXiv:1804.10540 [cs.LO], 2018.
%H A267827 Noam Zeilberger, <a href="https://arxiv.org/abs/1803.10080">A Sequent Calculus for a Semi-Associative Law</a>, arXiv preprint 1803.10030, March 2018 (A revised version of a 2017 conference paper)
%H A267827 Noam Zeilberger, <a href="https://vimeo.com/289907363">A proof-theoretic analysis of the rotation lattice of binary trees, Part 1 (video)</a>, Rutgers Experimental Math Seminar, Sep 13 2018. Part 2 is vimeo.com/289910554.
%H A267827 Noam Zeilberger, <a href="http://noamz.org/talks/expmath.2020.06.18.pdf">From Lambda Calculus to the Four Color Theorem, via Experimental Mathematics (slides)</a>, Rutgers Experimental Math Seminar, Jun 18 2020. For the video see http://noamz.org/videos/expmath.2020.06.18.mp4.
%H A267827 Noam Zeilberger, <a href="/A267827/a267827.pdf">From Lambda Calculus to the Four Color Theorem, via Experimental Mathematics (slides)</a>, Rutgers Experimental Math Seminar, Jun 18 2020. [Local copy]
%F A267827 The o.g.f. f(z) = z + 2*z^3 + 20*z^5 + 352*z^7 + ... can be defined using a catalytic variable as f(z) = F(z,0), where F(z,x) satisfies the functional-differential equation F(z,x) = x + z*(F(z,x) - F(z,0))^2 + z*(d/dx)F(z,x).
%F A267827 From _Gheorghe Coserea_, Nov 10 2017: (Start)
%F A267827 0 = x^5*y*y' + y - x^2, where y(x) = x^2*A(-x^6).
%F A267827 0 = 6*y*y'*x^2 + 2*y^2*x - y + 1, where y(x) = A(x).
%F A267827 a(n) = (6*n-2)*a(n-1) + Sum_{k=1..n-2} (6*k+2)*a(k)*a(n-1-k), for n >= 2.
%F A267827 (End)
%F A267827 a(n) = A291843(3*n+1, 2*n), n >= 1. - _Danny Rorabaugh_, Nov 10 2017
%e A267827 A(x) = 1 + 2*x + 20*x^2 + 352*x^3 + 8624*x^4 + 266784*x^5 + ...
%t A267827 a[0] = 1; a[1] = 2; a[n_] := a[n] = (6n-2) a[n-1] + Sum[(6k+2) a[k] a[n-1-k], {k, 1, n-2}];
%t A267827 Table[a[n], {n, 0, 16}] (* _Jean-François Alcover_, Oct 16 2018, after _Gheorghe Coserea_ *)
%o A267827 (PARI)
%o A267827 seq(N) = {
%o A267827   my(a = vector(N)); a[1] = 2;
%o A267827   for(n=2, N,
%o A267827     a[n] = (6*n-2)*a[n-1] + sum(k=1, n-2, (6*k+2)*a[k]*a[n-1-k]));
%o A267827   concat(1,a);
%o A267827 };
%o A267827 seq(16)
%o A267827 \\ test 1: y = x^2*subst(Ser(seq(201)),'x,-'x^6); 0 == x^5*y*y' + y - x^2
%o A267827 \\ test 2: y = Ser(seq(201)); 0 == 6*y*y'*x^2 + 2*y^2*x - y + 1
%o A267827 \\ _Gheorghe Coserea_, Nov 10 2017
%o A267827 F(N) = {
%o A267827   my(x='x+O('x^N), t='t, F0=x, F1=0, n=1);
%o A267827   while(n++,
%o A267827     F1 = t + x*(F0 - subst(F0,t,0))^2 + x*deriv(F0,t);
%o A267827     if (F1 == F0, break()); F0 = F1;);
%o A267827   F0;
%o A267827 };
%o A267827 seq(N) = my(v=Vec(subst(F(2*N+2),'t,0))); vector((#v+1)\2, n, v[2*n-1]);
%o A267827 seq(16) \\ _Gheorghe Coserea_, Apr 01 2017
%Y A267827 Sequences mentioned in the Noam Zeilberger 2018 video: A000168, A000260, A000309, A000698, A000699, A002005, A062980, A267827.
%Y A267827 Cf. A000309, A062980.
%K A267827 nonn
%O A267827 0,2
%A A267827 _Noam Zeilberger_, Jan 21 2016