OFFSET
0,2
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..1000
Maciej Bendkowski, Katarzyna Grygiel, Pierre Lescanne, and Marek Zaionc, Combinatorics of λ-terms: a natural approach, arXiv:1609.08106 [cs.LO], 2016.
Maciej Bendkowski, Katarzyna Grygiel, Pierre Lescanne, and Marek Zaionc, A Natural Counting of Lambda Terms, arXiv preprint arXiv:1506.02367 [cs.LO], 2015.
Maciej Bendkowski and Pierre Lescanne, On the enumeration of closures and environments with an application to random generation, Logical Methods in Computer Science (2019) Vol. 15, No. 4, 3:1-3:21.
K. Grygiel and P. Lescanne, A natural counting of lambda terms, Preprint 2015.
FORMULA
G.f. G(z) satisfies z*G(z)^2 - (1-z)*G(z) + 1/(1-z) = 0 (see Bendkowski link Appendix B, p. 23). - Michel Marcus, Jun 30 2015
a(n) ~ 3^(n+1/2) * sqrt(43/(2*((43*(3397 - 261*sqrt(129)))^(1/3) + (43*(3397 + 261*sqrt(129)))^(1/3) - 86)*Pi)) / (3 - (2*6^(2/3)) / (sqrt(129)-9)^(1/3) + (6*(sqrt(129)-9))^(1/3))^n / (2*n^(3/2)). - Vaclav Kotesovec, Jul 01 2015
a(n) = 1 + a(n-1) + Sum_{i=0..n-1} a(i)*a(n-1-i). - Vladimir Kruchinin, May 03 2018
a(n) = Sum_{i=0..n} Sum_{k=1..n-i} binomial(k+i-1,k-1)*binomial(2*k+i-2,k+i-1)*binomial(n-i-1,n-k-i)/k. - Vladimir Kruchinin, May 03 2018
a(n) = Sum_{i=0..n-1} hypergeom([(i+1)/2, i/2+1, i-n+1], [1, 2], -4). - Peter Luschny, May 03 2018
From Peter Bala, Sep 02 2024: (Start)
a(n) = Sum_{k = 0..n} 1/(k + 1) * binomial(2*k, k)*binomial(n+2*k+1, 3*k+1).
a(n) = (n + 1)*hypergeom([1/2, -n, (n+2)/2, (n+3)/2], [2, 2/3, 4/3], -16/27).
P-recursive: (n + 1)*a(n) = (8*n - 3)*a(n-1) - (10*n - 13)*a(n-2) + (4*n - 11)*a(n-3) - (n - 4)*a(n-4) with a(0) = 1, a(1) = 3, a(2) = 10 and a(3) = 40.
G.f. A(x) = 1/(1 - x)^2 * c(x/(1-x)^3) = (1 - x - sqrt((1 - 7*x + 3*x^2 - x^3)/(1 - x)))/(2*x), where c(x) = (1 - sqrt(1 - 4*x))/(2*x) is the g.f. of the Catalan numbers A000108. (End)
MAPLE
a:= proc(n) option remember; `if`(n<4, [1, 3, 10, 40][n+1],
((8*n-3)*a(n-1)-(10*n-13)*a(n-2)
+(4*n-11)*a(n-3)-(n-4)*a(n-4))/(n+1))
end:
seq(a(n), n=0..25); # Alois P. Heinz, Jun 30 2015
a := n -> add(hypergeom([(i+1)/2, i/2+1, i-n+1], [1, 2], -4), i=0..n-1):
seq(simplify(a(n)), n=0..25); # Peter Luschny, May 03 2018
MATHEMATICA
a[n_] := a[n] = If[n<4, {1, 3, 10, 40}[[n+1]], ((8*n-3)*a[n-1] - (10*n-13)*a[n-2] + (4*n-11)*a[n-3] - (n-4)*a[n-4])/(n+1)]; Table[a[n], {n, 0, 25}] (* Jean-François Alcover, Jul 22 2015, after Alois P. Heinz *)
PROG
(PARI) lista(nn) = {z = y + O(y^nn); Vec(((1-z)^2 - sqrt((1-z)^4-4*z*(1-z))) / (2*z*(1-z))); } \\ Michel Marcus, Jun 30 2015
(Maxima)
a(n):=sum(sum((binomial(k+i-1, k-1)*binomial(2*k+i-2, k+i-1)*binomial(n-i-1, n-k-i))/k, k, 1, n-i), i, 0, n); /* Vladimir Kruchinin, May 03 2018 */
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Kellen Myers, Jun 15 2015
EXTENSIONS
More terms from Michel Marcus, Jun 30 2015
STATUS
approved