A097472 Number of different candle trees having a total of m edges.

1, 3, 10, 31, 96, 296, 912, 2809, 8651, 26642, 82047, 252672, 778128, 2396320, 7379697, 22726483, 69988378, 215535903, 663763424, 2044122936, 6295072048, 19386276329, 59701891739, 183857684514, 566207320575, 1743689586432

Offset: 0

Author

Alexander Malkis, Sep 18 2004

Keywords

• easy
• nice
• nonn

Comments

A candle tree is a graph on the plane square lattice Z X Z whose edges have length one with the following properties: (a) It contains a line segment ("trunk") of length from 0 to m on the vertical axis, its lowest node is at the origin. (b) It contains horizontal line segments ("branches"); each of them intersects the trunk. (c) Each branch is allowed to have "candles", which are vertical edges of length 1, whose lower node is on a branch.

Row sums of triangle in A238241. - Philippe Deléham, Feb 21 2014

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..200
• Alexander Malkis, Polyedges, polyominoes and the 'Digit' game, diploma thesis in computer science, Universität des Saarlandes, 2003, Saarbrücken.
• Index entries for linear recurrences with constant coefficients, signature (3,1,-2,-1).

Crossrefs

Bisection of A060945 and |A077930|.

Programs

Mathematica

CoefficientList[Series[1/(x^4+2x^3-x^2-3x+1),{x,0,30}],x] (* or *) LinearRecurrence[{3,1,-2,-1},{1,3,10,31},30] (* Harvey P. Dale, Jun 14 2011 *)

Maxima

a(n):=sum(sum(binomial(k,n-m-k+1)*binomial(k+2*m-1,2*m-1),k,1,n-m+1),m,1,n)+1; /* Vladimir Kruchinin, May 12 2011 */

PARI

a(n)=sum(m=1,n,sum(k=1,n-m+1,binomial(k,n-m-k+1)*binomial(k+2*m-1,2*m-1))) \\ Charles R Greathouse IV, Jun 17 2013

Formula

a(n) = Sum_{s, d, k>=0 with s+d+k=m} binomial(s+2d+1, s)*binomial(s, k);

generating function = 1/((1-x)*(1-2*x-3*x^2-x^3)).

a(n) = 3*a(n-1) + a(n-2) - 2*a(n-3) - a(n-4);

a(n) = 1 + Sum_{m=1..n} Sum_{k=1..n-m+1} binomial(k, n-m-k+1)*binomial(k+2*m-1,2*m-1). - Vladimir Kruchinin, May 12 2011

a(n) = Sum_{k=0..n} A238241(n,k). - Philippe Deléham, Feb 21 2014

a(n) - a(n-1) = A218836(n). - R. J. Mathar, Jun 17 2020