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A271941 Number of peaks in all bargraphs having semiperimeter n (n>=2).

%I A271941 #14 Jun 26 2022 22:46:42
%S A271941 1,2,5,13,36,105,317,979,3070,9731,31090,99940,322832,1047007,3407017,
%T A271941 11118165,36370984,119234791,391620238,1288394790,4244993865,
%U A271941 14005026856,46260856498,152974164616,506355410344,1677603452621,5562725698010,18459595624048,61301038293810,203705244194997
%N A271941 Number of peaks in all bargraphs having semiperimeter n (n>=2).
%H A271941 Robert Israel, <a href="/A271941/b271941.txt">Table of n, a(n) for n = 2..1880</a>
%H A271941 A. Blecher, C. Brennan, and A. Knopfmacher, <a href="http://dx.doi.org/10.1080/0035919X.2015.1059905">Peaks in bargraphs</a>, Trans. Royal Soc. South Africa, 71, No. 1, 2016, 97-103.
%H A271941 M. Bousquet-Mélou and A. Rechnitzer, <a href="http://dx.doi.org/10.1016/S0196-8858(02)00553-5">The site-perimeter of bargraphs</a>, Adv. in Appl. Math. 31 (2003), 86-112.
%F A271941 a(n) = Sum_{k>=1} k*A271940(n,k).
%F A271941 G.f.: z^2*((1+z^2)*sqrt(1-4z+2z^2+z^4)+1-4z+2z^2+z^4)/(2(1-3z-z^2-z^3)(1-z)^2).
%F A271941 (1-n)*a(n)-a(n+1)+(-4-3*n)*a(n+2)+(-2+4*n)*a(n+3)+(-9-3*n)*a(4+n)+(15+4*n)*a(n+5)+(-4-n)*a(n+6)+2 = 0. - _Robert Israel_, May 20 2016
%e A271941 a(4)=5 because each of the 5 (=A082582(4)) bargraphs of semiperimeter 4 (corresponding to the compositions [1,1,1],[1,2],[2,1],[2,2],[3]) has only 1 peak.
%e A271941 a(6)=36 because among the 35 (=A082582(6)) bargraphs of semiperimeter 6 only the one corresponding to the composition [2,1,2] has 2 peaks; 34*1 + 1* 2 = 36.
%p A271941 g := (1/2)*z^2*((1+z^2)*sqrt(1-4*z+2*z^2+z^4)+1-4*z+2*z^2+z^4)/((1-z)^2*(1-3*z-z^2-z^3)): gser := series(g, z = 0, 40): seq(coeff(gser, z, n), n = 2 .. 35);
%Y A271941 Cf. A082582, A271940.
%Y A271941 Partial sums of A271941.
%K A271941 nonn
%O A271941 2,2
%A A271941 _Emeric Deutsch_, May 20 2016