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A213705 a(n)=n if n <= 3, otherwise a(n) = A007477(n-1)+A007477(n). 2
1, 2, 3, 5, 9, 17, 33, 66, 134, 277, 579, 1224, 2610, 5609, 12135, 26408, 57770, 126962, 280192, 620674, 1379586, 3075943, 6877611, 15417934, 34646156, 78027146, 176087292, 398143230, 901827322, 2046112299, 4649558191, 10581041518, 24112473412, 55019560650, 125696393844, 287494670302 (list; graph; refs; listen; history; text; internal format)
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.

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

(Scheme): (define (A213705 n) (if (< n 2) n (+ (A007477 (- n 1)) (A007477 n))))

(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 xrange(n - 1)])

def a(n): return n if n<3 else b(n - 1) + b(n)

print [a(n) for n in xrange(1, 51)] # Indranil Ghosh, Apr 11 2017

CROSSREFS

Sequence in context: A248155 A000051 A094373 * A061902 A166286 A179807

Adjacent sequences:  A213702 A213703 A213704 * A213706 A213707 A213708

KEYWORD

nonn,easy

AUTHOR

Antti Karttunen, Sep 14 2012

STATUS

approved

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Last modified August 21 19:52 EDT 2017. Contains 290906 sequences.