A273343 The sum of the first-column lengths of all bargraphs of semiperimeter n (n>=2).

1, 3, 9, 26, 75, 218, 640, 1898, 5682, 17155, 52187, 159827, 492417, 1525222, 4746906, 14837444, 46558573, 146614539, 463186317, 1467631144, 4662899110, 14851847390, 47414162252, 151692982789, 486280700344, 1561757802585, 5024492606869, 16191028967145, 52253656263073, 168880350860512

Offset

2,2

Links

Table of n, a(n) for n=2..31.

M. Bousquet-Mélou and A. Rechnitzer, The site-perimeter of bargraphs, Adv. in Appl. Math. 31 (2003), 86-112.

Emeric Deutsch, S Elizalde, Statistics on bargraphs viewed as cornerless Motzkin paths, arXiv preprint arXiv:1609.00088, 2016

Formula

G.f.: G = (z^6 + z^5 + z^4 - 2z^3 - 5z^2 +5z - 1 - (z^3 + 2z^2 + 2z - 1)Z)/(2 Z z^2), where Z = (1 - z)sqrt((1 - z)(1 - 3z - z^2 - z^3)).

a(n) = Sum(k*A273342(n,k), k>=1).

Conjecture: (n+2)*a(n) +(-5*n-3)*a(n-1) +(5*n-2)*a(n-2) +(2*n-9)*a(n-3) +(-n+3)*a(n-4) +(-n+4)*a(n-5) +(-n+7)*a(n-6)=0. - R. J. Mathar, Jun 02 2016

Example

a(4)=9  because the 5 (=A082582(4)) bargraphs of semiperimeter 4 correspond to the compositions [1,1,1], [1,2], [2,1], [2,2], [3] and, clearly, the sum of their first-columns lengths is 1+1+2+2+3=9.

Maple

Z := (1-z)*sqrt((1-z)*(1-3*z-z^2-z^3)): g := ((z^6+z^5+z^4-2*z^3-5*z^2+5*z-1-(z^3+2*z^2+2*z-1)*Z)*(1/2))/(z^2*Z): gser := series(g, z=0, 44): seq(coeff(gser, z, n), n=2..40);

Mathematica

Drop[CoefficientList[Series[(x^6 + x^5 + x^4 - 2 x^3 - 5 x^2 + 5 x - 1 - (x^3 + 2 x^2 + 2 x - 1) #)/(2 # x^2) &[(1 - x) Sqrt[(1 - x) (1 - 3 x - x^2 - x^3)]], {x, 0, 31}], x], 2] (* Michael De Vlieger, May 21 2016 *)

Crossrefs

Cf. A082582, A273342.

Sequence in context: A018919 A123941 A005774 * A101169 A119826 A027915

Adjacent sequences: A273340 A273341 A273342 * A273344 A273345 A273346

Keyword

nonn

Author

Emeric Deutsch, May 21 2016

Status

approved