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