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A025265
a(n) = a(1)*a(n-1) + a(2)*a(n-2) + ... + a(n-1)*a(1) for n >= 4.
3
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
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
Sequence in context: A055588 A088456 A091561 * A152225 A037245 A244886
KEYWORD
nonn
STATUS
approved