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A258973 The number of plain lambda terms presented by de Bruijn indices, see Bendkowski et al. where zeros have no weight. 2
1, 3, 10, 40, 181, 884, 4539, 24142, 131821, 734577, 4160626, 23881695, 138610418, 812104884, 4796598619, 28529555072, 170733683579, 1027293807083, 6211002743144, 37713907549066, 229894166951757, 1406310771154682, 8630254073158599, 53117142215866687, 327800429456036588 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

LINKS

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Maciej Bendkowski, Katarzyna Grygiel, Pierre Lescanne, Marek Zaionc, Combinatorics of λ-terms: a natural approach, arXiv:1609.08106 [cs.LO], 2016.

Maciej Bendkowski, Katarzyna Grygiel, Pierre Lescanne, Marek Zaionc, A Natural Counting of Lambda Terms, arXiv preprint arXiv:1506.02367 [cs.LO], 2015.

K. Grygiel, 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

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

Cf. A105633, A114851.

Sequence in context: A151076 A151077 A300043 * A217885 A216367 A003703

Adjacent sequences:  A258970 A258971 A258972 * A258974 A258975 A258976

KEYWORD

nonn

AUTHOR

Kellen Myers, Jun 15 2015

EXTENSIONS

More terms from Michel Marcus, Jun 30 2015

STATUS

approved

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Last modified May 19 02:45 EDT 2019. Contains 323377 sequences. (Running on oeis4.)