A106597 -id:A106597 - cp's OEIS frontend

A118649 Row sums for A106597.

1, 2, 5, 12, 30, 74, 184, 456, 1132, 2808, 6968, 17288, 42896, 106432, 264080, 655232, 1625760, 4033824, 10008704, 24833536, 61616832, 152883328, 379333248, 941199488, 2335298816, 5794330112, 14376858880, 35671780352, 88508618240

Offset: 0

Joshua Zucker, May 10 2006

nonn

For n>=2, a(n)= Number of "stable LEGO walls" (i.e., walls in which seams don't match up from one level to the next) of width 7 and height n-1 when using bricks of length 2, 3, and 4. For example, there are a(2)=5 stable LEGO walls of height 1 and they are {[2,2,3],[2,3,2],[3,2,2],[3,4],[4,3]}. [Emilie Hogan, Oct 16 2009]
From Sean A. Irvine, Jun 05 2025: (Start)
Also, the number of walks of length n starting at vertex 2 in the graph:
    1-2
   /| |
  0 | |
   \| |
    4-3. (End)

Sean A. Irvine, Walks on Graphs.
S. Kitaev and J. Remmel, (a,b)-rectangle patterns in permutations and words, arXiv:1304.4286 [math.CO], 2013.

Cf. A106597.

Total[CoefficientList[#, y]]& /@ CoefficientList[(1 - x^2 y)/(1 - x - x y - 2 x^2 y + x^3 y + x^3 y^2) + O[x]^29, x] (* Jean-François Alcover, Oct 30 2018, after Emanuele Munarini in A106597 *)

From R. J. Mathar, Nov 10 2009: (Start)
a(n) = 2*a(n-1) + 2*a(n-2) - 2*a(n-3).
G.f.: (1-x^2) / (1-2*x-2*x^2+2*x^3). (End)

A118650 Center column a(2n,n) of A106597.

Original entry on oeis.org

1, 3, 14, 72, 388, 2150, 12134, 69370, 400398, 2328162, 13616892, 80022612, 472133638, 2794893246, 16592160944, 98743919468, 588913687966, 3518968100982, 21062444053040, 126256612255260, 757853797478188

Offset: 0

Joshua Zucker, May 10 2006

nonn

Number of lattice paths from (0,0) to (n,n) using steps (1,0), (0,1), and (s,s) for s>=1. [Joerg Arndt, Jul 01 2011]

Cf. A106597.

Table[Sum[Binomial[2k,k](-1)^(n-k)Sum[Binomial[k,j]Binomial[k+1,n-k-j]2^(k-j),{j,0,k}],{k,0,n}],{n,0,40}] (* Emanuele Munarini, Feb 06 2017 *)

makelist(sum(binomial(2*k,k)*(-1)^(n-k)*sum(binomial(k,j)*binomial(k+1,n-k-j)*2^(k-j),j,0,k),k,0,n),n,0,12); /* Emanuele Munarini, Feb 06 2017 */

G.f.: (1-x)/sqrt(1-8*x+12*x^2-4*x^3). - Mark van Hoeij, Apr 16 2013
Conjecture: n*a(n) +3*(-3*n+2)*a(n-1) +4*(5*n-8)*a(n-2) +2*(-8*n+21)*a(n-3) +2*(2*n-7)*a(n-4)=0. - R. J. Mathar, Nov 10 2013
From Emanuele Munarini, Feb 06 2017: (Start)
a(n) = Sum_{k=0..n} binomial(2*k,k)*(-1)^(n-k) * Sum_{j=0..k} binomial(k,j)*binomial(k+1,n-k-j)*2^(k-j).
Proof of Mathar's recurrence.
Let A(t) be the g.f. of the coefficients a(n). Then we have the identity (1 - 9*t + 20*t^2 - 16*t^3 + 4*t^4)*A'(t) = (3 - 8*t + 6*t^2 - 2*t^3)*A(t).
Let R be the incremental ratio, i.e. the operator defined by RA(t) = (A(t)-A(0))/t, giving the g.f. of the shifted sequence a(n+1), then we have
R^3A'(t) - 9*R^2A'(t) + 20*RA'(t) - 16*A'(t) + 4*t*A'(t) = 3*R^3A(t) - 8*R^2A(t) + 6*RA(t) - 2*A(t), from which we obtain the recurrence:
(n+4)*a(n+4)-3*(3*n+10)*a(n+3)+4*(5*n+12)*a(n+2)-2*(8*n+11)*a(n+1)+2*(2*n+1)*a(n)=0.
This proves the above conjecture.
(End)

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A118649 Row sums for A106597.

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