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A267827
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Number of closed indecomposable linear lambda terms with 2n+1 applications and abstractions.
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14
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1, 2, 20, 352, 8624, 266784, 9896448, 426577920, 20918138624, 1149216540160, 69911382901760, 4665553152081920, 338942971881472000, 26631920159494995968, 2250690001888540950528, 203595258621775065120768, 19629810220331494121865216
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OFFSET
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0,2
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COMMENTS
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A linear lambda term is indecomposable if it has no closed proper subterm.
Equivalently, number of closed bridgeless rooted trivalent maps (on compact oriented surfaces of arbitrary genus) with 2n+1 trivalent vertices (and 1 univalent vertex).
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
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LINKS
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FORMULA
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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).
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)
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EXAMPLE
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A(x) = 1 + 2*x + 20*x^2 + 352*x^3 + 8624*x^4 + 266784*x^5 + ...
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MATHEMATICA
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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}];
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PROG
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(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
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]);
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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