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

1, 2, 5, 13, 36, 105, 317, 979, 3070, 9731, 31090, 99940, 322832, 1047007, 3407017, 11118165, 36370984, 119234791, 391620238, 1288394790, 4244993865, 14005026856, 46260856498, 152974164616, 506355410344, 1677603452621, 5562725698010, 18459595624048, 61301038293810, 203705244194997

Offset: 2

Author

Emeric Deutsch, May 20 2016

Keywords

• nonn

Examples

			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.
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.
		

Links

• Robert Israel, Table of n, a(n) for n = 2..1880
• A. Blecher, C. Brennan, and A. Knopfmacher, Peaks in bargraphs, Trans. Royal Soc. South Africa, 71, No. 1, 2016, 97-103.
• M. Bousquet-Mélou and A. Rechnitzer, The site-perimeter of bargraphs, Adv. in Appl. Math. 31 (2003), 86-112.

Crossrefs

Cf. A082582, A271940.

Partial sums of A271941.

Programs

Maple

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);

Formula

a(n) = Sum_{k>=1} k*A271940(n,k).

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).

(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