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Number of paths from (0,0) to (n,0), never going below the x-axis, using steps U=(1,1), H=(1,0) and D=(1,-1), where the H steps come in five colors.
8

%I #51 Feb 05 2024 10:43:13

%S 1,5,26,140,777,4425,25755,152675,919139,5606255,34578292,215322310,

%T 1351978807,8550394455,54419811354,348309105300,2240486766555,

%U 14476490777175,93914850905862,611489638708140,3994697746533171,26175407271617955,171991872078871311

%N Number of paths from (0,0) to (n,0), never going below the x-axis, using steps U=(1,1), H=(1,0) and D=(1,-1), where the H steps come in five colors.

%C Number of 3-colored Schroeder paths from (0,0) to (2n+2,0) with no level steps H=(2,0) at even level. H-steps at odd levels are colored with one of the three colors. Example: a(2)=5 because we have UUDD, UHD (3 choices) and UDUD. - _José Luis Ramírez Ramírez_, Apr 27 2015

%H Vincenzo Librandi, <a href="/A182401/b182401.txt">Table of n, a(n) for n = 0..200</a>

%F a(n) = [x^n] (1+5*x+x^2)^(n+1)/(n+1).

%F a(n) = Sum_{k=0..floor(n/2)} (binomial(n,2*k)*binomial(2*k,k)/(k+1))*5^(n-2*k).

%F G.f.: (1-5*x-sqrt(1-10*x+21*x^2))/(2*x^2).

%F Conjecture: (n+2)*a(n) +5*(-2*n-1)*a(n-1) +21*(n-1)*a(n-2)=0. - _R. J. Mathar_, Jul 24 2012

%F a(n) ~ 7^(n+3/2)/(2*sqrt(Pi)*n^(3/2)). - _Vaclav Kotesovec_, Oct 20 2012

%F a(n) = A125906(n,0). - _Philippe Deléham_, Mar 04 2013

%F G.f.: 1/(1 - 5*x - x^2/(1 - 5*x - x^2/(1 - 5*x - x^2/(1 - 5*x - x^2/(1 - ...))))), a continued fraction. - _Ilya Gutkovskiy_, Sep 21 2017

%F From _Seiichi Manyama_, Jan 15 2024: (Start)

%F G.f.: (1/x) * Series_Reversion( x / (1+5*x+x^2) ).

%F a(n) = (1/(n+1)) * Sum_{k=0..n} 3^(n-k) * binomial(n+1,n-k) * binomial(2*k+2,k). (End)

%F From _Peter Bala_, Feb 03 2024: (Start)

%F G.f: 1/(1 - 3*x)*c(x/(1 - 3*x))^2 = 1/(1 - 7*x)*c(-x/(1 - 7*x))^2, where c(x) = (1 - sqrt(1 - 4*x))/(2*x) is the g.f. of the Catalan numbers A000108.

%F a(n) = Sum_{k = 0..n} 3^(n-k)*binomial(n, k)*Catalan(k+1).

%F a(n) = 3^n * hypergeom([3/2, -n], [3], -4/3).

%F a(n) = 7^n * Sum_{k = 0..n} (-7)^(-k)*binomial(n, k)*Catalan(k+1).

%F a(n) = 7^n * hypergeom([3/2, -n], [3], 4/7). (End)

%e seq(3^n * simplify(hypergeom([3/2, -n], [3], -4/3)), n = 0..20); # _Peter Bala_, Feb 04 2024

%t CoefficientList[Series[(1-5*x-Sqrt[1-10*x+21*x^2])/(2*x^2), {x, 0, 20}], x] (* _Vaclav Kotesovec_, Oct 20 2012 *)

%t a[n_] := 5^n*Hypergeometric2F1[(1-n)/2, -n/2, 2, 4/25]; Table[a[n], {n, 0, 22}] (* _Jean-François Alcover_, Feb 22 2013, after 2nd formula *)

%o (Maxima) a(n):=coeff(expand((1+5*x+x^2)^(n+1)),x^n)/(n+1);

%o makelist(a(n),n,0,30);

%o (PARI) x='x+O('x^66); Vec((1-5*x-sqrt(1-10*x+21*x^2))/(2*x^2)) \\ _Joerg Arndt_, Jun 02 2013

%Y Cf. A001006, A002212, A005572, A025230, A126120, A257290.

%K nonn,easy

%O 0,2

%A _Emanuele Munarini_, Apr 27 2012