A267827 Number of closed indecomposable linear lambda terms with 2n+1 applications and abstractions.
1, 2, 20, 352, 8624, 266784, 9896448, 426577920, 20918138624, 1149216540160, 69911382901760, 4665553152081920, 338942971881472000, 26631920159494995968, 2250690001888540950528, 203595258621775065120768, 19629810220331494121865216
Offset: 0
Keywords
Examples
A(x) = 1 + 2*x + 20*x^2 + 352*x^3 + 8624*x^4 + 266784*x^5 + ...
Links
- Gheorghe Coserea, Table of n, a(n) for n = 0..303
- Lawrence Dresner, Protection of a test magnet wound with a Ag/BSCCO high-temperature superconductor, Oak Ridge National Lab technical report (ORNL/HTSPC-3), 1992. See Eq. (25).
- Noam Zeilberger, Linear lambda terms as invariants of rooted trivalent maps, 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, 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), Rutgers Experimental Math Seminar, Sep 13 2018. Part 2 is vimeo.com/289910554.
- Noam Zeilberger, From Lambda Calculus to the Four Color Theorem, via Experimental Mathematics (slides), Rutgers Experimental Math Seminar, Jun 18 2020. For the video see http://noamz.org/videos/expmath.2020.06.18.mp4.
- Noam Zeilberger, From Lambda Calculus to the Four Color Theorem, via Experimental Mathematics (slides), Rutgers Experimental Math Seminar, Jun 18 2020. [Local copy]
Crossrefs
Programs
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Mathematica
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}]; Table[a[n], {n, 0, 16}] (* Jean-François Alcover, Oct 16 2018, after Gheorghe Coserea *) -
PARI
seq(N) = { my(a = vector(N)); a[1] = 2; for(n=2, N, a[n] = (6*n-2)*a[n-1] + sum(k=1, n-2, (6*k+2)*a[k]*a[n-1-k])); concat(1,a); }; seq(16) \\ test 1: y = x^2*subst(Ser(seq(201)),'x,-'x^6); 0 == x^5*y*y' + y - x^2 \\ test 2: y = Ser(seq(201)); 0 == 6*y*y'*x^2 + 2*y^2*x - y + 1 \\ Gheorghe Coserea, Nov 10 2017 F(N) = { my(x='x+O('x^N), t='t, F0=x, F1=0, n=1); while(n++, F1 = t + x*(F0 - subst(F0,t,0))^2 + x*deriv(F0,t); if (F1 == F0, break()); F0 = F1;); F0; }; seq(N) = my(v=Vec(subst(F(2*N+2),'t,0))); vector((#v+1)\2, n, v[2*n-1]); seq(16) \\ Gheorghe Coserea, Apr 01 2017
Formula
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).
From Gheorghe Coserea, Nov 10 2017: (Start)
0 = x^5*y*y' + y - x^2, where y(x) = x^2*A(-x^6).
0 = 6*y*y'*x^2 + 2*y^2*x - y + 1, where y(x) = A(x).
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.
(End)
a(n) = A291843(3*n+1, 2*n), n >= 1. - Danny Rorabaugh, Nov 10 2017
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