A253831 Number of 2-Motzkin paths with no level steps at height 1.
1, 2, 5, 12, 30, 76, 197, 522, 1418, 3956, 11354, 33554, 102104, 319608, 1027237, 3381714, 11371366, 38946892, 135505958, 477781296, 1703671604, 6132978608, 22256615602, 81327116484, 298938112816, 1104473254912, 4098996843500, 15272792557230, 57106723430892, 214202598271360, 805743355591301
Offset: 0
Keywords
Links
- Robert Israel, Table of n, a(n) for n = 0..1000
Programs
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Maple
rec:= (54+36*n)*a(n)+(-3+7*n)*a(n+1)+(-60-36*n)*a(n+2)+(36+16*n)*a(n+3)+(-6-2*n)*a(n+4) = 0: f:= gfun:-rectoproc({rec,seq(a(i)=[1,2,5,12][i+1],i=0..3)},a(n),remember): seq(f(n),n=0..100); # Robert Israel, Apr 29 2015 -
Mathematica
CoefficientList[Series[1/(1-2*x-x*((1-Sqrt[1-4*x])/(3-Sqrt[1-4*x]))), {x, 0, 20}], x] (* Vaclav Kotesovec, Apr 21 2015 *) -
Maxima
a(n):=sum(sum(((sum((k+1)*binomial(k+m,k+1)*binomial(2*j-k+m-1,j-k)*(-1)^(k),k,0,j))*2^(n-j-2*m)*binomial(n-m-j,m))/(j+m),j,0,n-2*m),m,1,n/2)+2^n; /* Vladimir Kruchinin, Mar 11 2016 */
Formula
G.f.: 1/(1-2*x-x*F(x)), where F(x) is the g.f. of Fine numbers A000957.
G.f.: 2*(2+x)/(4-7*x-6*x^2+x*sqrt(1-4*x)).
a(n) ~ 4^(n+1) / (25*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Apr 21 2015
(54+36*n)*a(n)+(-3+7*n)*a(n+1)+(-60-36*n)*a(n+2)+(36+16*n)*a(n+3)+(-6-2*n)*a(n+4) = 0. - Robert Israel, Apr 29 2015
a(n) = Sum_{m=1..n/2}(Sum_{j=0..n-2*m}(((Sum_{k=0..j}((k+1)*binomial(k+m,k+1)*binomial(2*j-k+m-1,j-k)*(-1)^(k)))*2^(n-j-2*m)*binomial(n-m-j,m))/(j+m)))+2^n. - Vladimir Kruchinin, Mar 11 2016
Comments