A025265 - OEIS

%I #49 Nov 23 2024 11:12:38

%S 1,0,1,2,4,9,22,56,146,388,1048,2869,7942,22192,62510,177308,506008,

%T 1451866,4185788,12119696,35227748,102753800,300672368,882373261,

%U 2596389190,7658677856,22642421206,67081765932,199128719896,592179010350,1764044315540

%N a(n) = a(1)*a(n-1) + a(2)*a(n-2) + ... + a(n-1)*a(1) for n >= 4.

%C With offset 0, a(n) is the number of 021-avoiding ascent sequences of length n with no isolated 0's. For example, a(4)=4 counts 0000, 0001, 0011, 0012. - _David Callan_, Nov 13 2019

%H G. C. Greubel, <a href="/A025265/b025265.txt">Table of n, a(n) for n = 1..1000</a>

%H Jean-Luc Baril, Rigoberto Flórez, and José L. Ramírez, <a href="http://jl.baril.u-bourgogne.fr/symasympyramid.pdf">Counting symmetric and asymmetric peaks in motzkin paths with air pockets</a>, Univ. Bourgogne (France, 2023).

%H Paul Barry, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL19/Barry/barry321.html">Jacobsthal Decompositions of Pascal's Triangle, Ternary Trees, and Alternating Sign Matrices</a>, Journal of Integer Sequences, 19, 2016, #16.3.5.

%H David Callan, <a href="https://arxiv.org/abs/1911.02209">On Ascent, Repetition and Descent Sequences</a>, arXiv:1911.02209 [math.CO], 2019.

%F a(n+2) = A091561(n).

%F G.f.: (1-sqrt(1-4*x+4*x^2-4*x^3))/2. - _Michael Somos_, Jun 08 2000

%F G.f. A(x) satisfies 0=f(x, A(x)) where f(x, y)=(x-x^2+x^3)-(y-y^2). - _Michael Somos_, May 26 2005

%F D-finite with recurrence n*a(n) +2*(3-2*n)*a(n-1) +4*(n-3)*a(n-2)+ 2*(9-2*n)*a(n-3)=0. - _R. J. Mathar_, Aug 14 2012

%F a(n) = Sum_{k=0..n} C(k)*Sum_{j=0..k+1} binomial(j,n-k-j-1)*binomial(k+1,j)*(-1)^(-n+k-1), where C(k) is Catalan numbers (A000108)  - _Vladimir Kruchinin_, May 10 2018

%t nmax = 30; aa = ConstantArray[0,nmax]; aa[[1]] = 1; aa[[2]] = 0; aa[[3]] = 1; Do[aa[[n]] = Sum[aa[[k]] * aa[[n-k]],{k,1,n-1}],{n,4,nmax}]; aa (* _Vaclav Kotesovec_, Jan 25 2015 *)

%t CoefficientList[Series[(1-Sqrt[1-4x+4x^2-4x^3])/2,{x,0,40}],x] (* _Harvey P. Dale_, Jun 02 2017 *)

%o (PARI) a(n)=polcoeff((1-sqrt(1-4*x+4*x^2-4*x^3+x*O(x^n)))/2,n)

%o (PARI) a(n)=if(n<1,0,polcoeff(subst(serreverse(x-x^2+x*O(x^n)),x,x-x^2+x^3),n))

%o (Maxima)

%o a(n):=sum((binomial(2*k,k)*(sum(binomial(j,n-k-j-1)*binomial(k+1,j),j,0,k+1))*(-1)^(-n+k+1))/(k+1),k,0,n); /* _Vladimir Kruchinin_, May 10 2018  */

%Y Cf. A025247, A091561.

%K nonn

%O 1,4

%A _Clark Kimberling_