A025265 a(n) = a(1)*a(n-1) + a(2)*a(n-2) + ... + a(n-1)*a(1) for n >= 4.

1, 0, 1, 2, 4, 9, 22, 56, 146, 388, 1048, 2869, 7942, 22192, 62510, 177308, 506008, 1451866, 4185788, 12119696, 35227748, 102753800, 300672368, 882373261, 2596389190, 7658677856, 22642421206, 67081765932, 199128719896, 592179010350, 1764044315540

Offset

1,4

Comments

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

Links

G. C. Greubel, Table of n, a(n) for n = 1..1000

Jean-Luc Baril, Rigoberto Flórez, and José L. Ramírez, Counting symmetric and asymmetric peaks in motzkin paths with air pockets, Univ. Bourgogne (France, 2023).

Paul Barry, Jacobsthal Decompositions of Pascal's Triangle, Ternary Trees, and Alternating Sign Matrices, Journal of Integer Sequences, 19, 2016, #16.3.5.

David Callan, On Ascent, Repetition and Descent Sequences, arXiv:1911.02209 [math.CO], 2019.

Formula

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

G.f.: (1-sqrt(1-4*x+4*x^2-4*x^3))/2. - Michael Somos, Jun 08 2000

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

Conjecture: 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

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

Mathematica

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

Prog

(PARI) a(n)=polcoeff((1-sqrt(1-4*x+4*x^2-4*x^3+x*O(x^n)))/2, n)
(PARI) a(n)=if(n<1, 0, polcoeff(subst(serreverse(x-x^2+x*O(x^n)), x, x-x^2+x^3), n))
(Maxima)
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  */

Crossrefs

Cf. A025247, A091561.

Sequence in context: A055588 A088456 A091561 * A152225 A037245 A244886

Adjacent sequences: A025262 A025263 A025264 * A025266 A025267 A025268

Keyword

nonn

Author

Clark Kimberling

Status

approved