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%I #50 Nov 17 2021 07:57:41
%S 1,2,3,5,9,17,33,66,134,277,579,1224,2610,5609,12135,26408,57770,
%T 126962,280192,620674,1379586,3075943,6877611,15417934,34646156,
%U 78027146,176087292,398143230,901827322,2046112299,4649558191,10581041518,24112473412,55019560650,125696393844,287494670302
%N a(n)=n if n <= 3, otherwise a(n) = A007477(n-1) + A007477(n).
%C 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.
%C See A007477 for the number of plausible parsings of "Buffalo^n" sentences in English.
%C 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.
%H Antti Karttunen, <a href="/A213705/b213705.txt">Table of n, a(n) for n = 1..1000</a>
%H Antti Karttunen, <a href="http://oeis.org/wiki/User:Antti_Karttunen/Etsiv%C3%A4t_etsiv%C3%A4t_etsiv%C3%A4t">Etsivät etsivät etsivät...</a>, OEIS Wiki.
%F Given the g.f. A(x) and the g.f. of A007853 B(x), then -x = A(-B(x)). - _Michael Somos_, Nov 07 2019
%e 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
%p b:= n-> coeff(series(RootOf(A=(A*x)^2+x+1, A), x, n+1), x, n):
%p a:= n-> `if`(n<2, n, b(n-1) +b(n)):
%p seq(a(n), n=1..40); # _Alois P. Heinz_, Sep 14 2012
%t (* 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 *)
%o (Scheme): (define (A213705 n) (if (< n 2) n (+ (A007477 (- n 1)) (A007477 n))))
%o (PARI) b(n) = sum(k=0, n - 2, binomial(2*k + 2, n - k - 2)*binomial(n - k - 2, k)/(k + 1));
%o a(n) = if(n<3, n, b(n - 1) + b(n)); \\ _Indranil Ghosh_, Apr 11 2017
%o (Python)
%o from sympy import binomial
%o 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)])
%o def a(n): return n if n<3 else b(n - 1) + b(n)
%o print([a(n) for n in range(1, 51)]) # _Indranil Ghosh_, Apr 11 2017
%o (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 */
%Y Cf. A007477, A007853.
%K nonn,easy
%O 1,2
%A _Antti Karttunen_, Sep 14 2012
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