%I #54 Feb 02 2022 16:11:23
%S 1,3,10,31,96,296,912,2809,8651,26642,82047,252672,778128,2396320,
%T 7379697,22726483,69988378,215535903,663763424,2044122936,6295072048,
%U 19386276329,59701891739,183857684514,566207320575,1743689586432
%N Number of different candle trees having a total of m edges.
%C 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.
%C Row sums of triangle in A238241. - _Philippe Deléham_, Feb 21 2014
%H Vincenzo Librandi, <a href="/A097472/b097472.txt">Table of n, a(n) for n = 0..200</a>
%H Alexander Malkis, <a href="https://wwwbroy.in.tum.de/~malkis/Malkis-dipl.pdf">Polyedges, polyominoes and the 'Digit' game</a>, diploma thesis in computer science, Universität des Saarlandes, 2003, Saarbrücken.
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (3,1,-2,-1).
%F a(n) = Sum_{s, d, k>=0 with s+d+k=m} binomial(s+2d+1, s)*binomial(s, k);
%F generating function = 1/((1-x)*(1-2*x-3*x^2-x^3)).
%F a(n) = 3*a(n-1) + a(n-2) - 2*a(n-3) - a(n-4);
%F 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
%F a(n) = Sum_{k=0..n} A238241(n,k). - _Philippe Deléham_, Feb 21 2014
%F a(n) - a(n-1) = A218836(n). - _R. J. Mathar_, Jun 17 2020
%t 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 *)
%o (Maxima)
%o 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 */
%o (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
%Y Bisection of A060945 and |A077930|.
%K easy,nice,nonn
%O 0,2
%A _Alexander Malkis_, Sep 18 2004