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 A253831 Number of 2-Motzkin paths with no level steps at height 1. 2
 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS For n=3 we have 12 paths: H(1)H(1)H(1), H(1)H(1)H(2), H(1)H(2)H(1), H(1)H(2)H(2), H(2)H(1)H(1), H(2)H(1)H(2), H(2)H(2)H(1), H(2)H(2)H(2), UDH(1), UDH(2), H(1)UD, H(2)UD. LINKS Robert Israel, Table of n, a(n) for n = 0..1000 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 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 *) PROG (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 */ CROSSREFS Cf. A000957, A217312, A257363. Sequence in context: A002026 A026938 A086622 * A024851 A188378 A145267 Adjacent sequences:  A253828 A253829 A253830 * A253832 A253833 A253834 KEYWORD nonn AUTHOR José Luis Ramírez Ramírez, Apr 20 2015 STATUS approved

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Last modified November 14 08:13 EST 2018. Contains 317174 sequences. (Running on oeis4.)