OFFSET
1,2
COMMENTS
a(n) gives the number of "plausible parsings" of the sentence "Etsivät^(n+1)" in Finnish (with the most common word order, SV & SVO), that is, sentences which consist only of n+1 copies of the word "etsivät". See the OEIS Wiki page.
See A007477 for the number of plausible parsings of "Buffalo^n" sentences in English.
In my view the value of a(0) should be 0 in this context (single word "Etsivät." is not a valid Finnish sentence, except as an answer to a question), although this is arguable. However, it is probably that this sequence occurs in other (combinatorial) contexts as well, and there a(0) might be something else than 0, so I left it off, and made the sequence start from offset 1.
LINKS
Antti Karttunen, Table of n, a(n) for n = 1..1000
Antti Karttunen, Etsivät etsivät etsivät..., OEIS Wiki.
FORMULA
Given the g.f. A(x) and the g.f. of A007853 B(x), then -x = A(-B(x)). - Michael Somos, Nov 07 2019
EXAMPLE
G.f. = x + 2*x^2 + 3*x^3 + 5*x^4 + 9*x^5 + 17*x^6 + 33*x^7 + ... - Michael Somos, Nov 07 2019
MAPLE
b:= n-> coeff(series(RootOf(A=(A*x)^2+x+1, A), x, n+1), x, n):
a:= n-> `if`(n<2, n, b(n-1) +b(n)):
seq(a(n), n=1..40); # Alois P. Heinz, Sep 14 2012
MATHEMATICA
(* b = A007477 *) b[n_] := Sum[Binomial[2*k+2, n-k-2]*Binomial[n-k-2, k]/(k + 1), {k, 0, n-2}]; a[n_] := b[n-1] + b[n]; a[1] = 1; a[2] = 2; Array[a, 40] (* Jean-François Alcover, Mar 04 2016 *)
PROG
(PARI) b(n) = sum(k=0, n - 2, binomial(2*k + 2, n - k - 2)*binomial(n - k - 2, k)/(k + 1));
a(n) = if(n<3, n, b(n - 1) + b(n)); \\ Indranil Ghosh, Apr 11 2017
(Python)
from sympy import binomial
def b(n): return sum([binomial(2*k + 2, n - k - 2)*binomial(n - k - 2, k)//(k + 1) for k in range(n - 1)])
def a(n): return n if n<3 else b(n - 1) + b(n)
print([a(n) for n in range(1, 51)]) # Indranil Ghosh, Apr 11 2017
(PARI) {a(n) = polcoeff( (1 + x) * (1 - 2*x^2 - sqrt(1 - 4*x^2 - 4*x^3 + x^3 * O(x^n))) / (2*x^2), n)}; /* Michael Somos, Nov 07 2019 */
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Antti Karttunen, Sep 14 2012
STATUS
approved