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A118430 Number of binary sequences of length n containing exactly one subsequence 010.

%I A118430 #19 Jul 18 2023 10:36:55
%S A118430 0,0,0,1,4,10,22,47,98,199,396,777,1508,2900,5534,10492,19782,37119,
%T A118430 69358,129118,239578,443229,817822,1505389,2764986,5068435,9273928,
%U A118430 16940488,30897020,56271128,102347564,185922589,337353688,611462514
%N A118430 Number of binary sequences of length n containing exactly one subsequence 010.
%C A118430 With only two 0's at the beginning, the convolution of A005314 with itself. Column 1 of A118429.
%H A118430 Alois P. Heinz, <a href="/A118430/b118430.txt">Table of n, a(n) for n = 0..1000</a>
%H A118430 T. Mansour and M. Shattuck, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL13/Shattuck/shattuck3.html">Counting Peaks and Valleys in a Partition of a Set</a>, J. Int. Seq. 13 (2010), 10.6.8, Lemma 2.1, k=2, 1 peak.
%H A118430 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (4,-6,6,-5,2,-1).
%F A118430 G.f.: z^3/(1-2*z+z^2-z^3)^2.
%e A118430 a(4) = 4 because we have 0100, 0101, 0010 and 1010.
%p A118430 g:=z^3/(1-2*z+z^2-z^3)^2: gser:=series(g,z=0,40): seq(coeff(gser,z,n),n=0..38);
%t A118430 LinearRecurrence[{4, -6, 6, -5, 2, -1}, {0, 0, 0, 1, 4, 10}, 40] (* _Jean-François Alcover_, May 11 2019 *)
%Y A118430 Cf. A005314, A118429, A255386.
%K A118430 nonn
%O A118430 0,5
%A A118430 _Emeric Deutsch_, Apr 27 2006