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 A076026 G.f.: (1-4*x*C)/(1-5*x*C) where C = (1/2-1/2*(1-4*x)^(1/2))/x = g.f. for Catalan numbers A000108. 7
 1, 1, 6, 37, 230, 1434, 8952, 55917, 349374, 2183230, 13643972, 85270626, 532926716, 3330739972, 20816939100, 130105200765, 813155081070, 5082210417270, 31763782696740, 198523522444950, 1240771573465140, 7754820693127020, 48467623215477120, 302922622226091090 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS a(n) is the number of Motzkin paths of length n-1 in which the (1,0)-steps at level 0 come in 6 colors and those at a higher level come in 2 colors. Example: a(4)=230 because, denoting  U=(1,1), H=(1,0), and D=(1,-1), we have 6^3 = 216 paths of shape HHH, 6 paths of shape HUD, 6 paths of shape UDH, and 2 paths of shape UHD. - Emeric Deutsch, May 02 2011 REFERENCES L. W. Shapiro and C. J. Wang, Generating identities via 2 X 2 matrices, Congressus Numerantium, 205 (2010), 33-46. LINKS FORMULA a(n+1)=Sum_{k, 0<=k<=n}A039598(n,k)*4^k . - Philippe Deléham, Mar 21 2007 a(n) = Sum_{k, 0<=k<=n}A039599(n,k)*A015521(k), for n>=1 . - Philippe Deléham, Nov 22 2007 Let A be the Toeplitz matrix of order n defined by: A[i,i-1]=-1, A[i,j]=Catalan(j-i), (i<=j), and A[i,j]=0, otherwise. Then, for n>=1, a(n+1)=(-1)^n*charpoly(A,-5). [From Milan Janjic, Jul 08 2010] From Gary W. Adamson, Jul 25 2011: (start) a(n) = upper left term in M^(n-1), M = an infinite square production matrix as follows: 6, 1, 0, 0, 0,... 1, 1, 1, 0, 0,... 1, 1, 1, 1, 0,... 1, 1, 1, 1, 1,... ... (end) Recurrence: 4*n*a(n) = (41*n-24)*a(n-1) - 50*(2*n-3)*a(n-2). - Vaclav Kotesovec, Dec 09 2013 a(n) ~ 3*5^(2*n-1)/4^(n+1). - Vaclav Kotesovec, Dec 09 2013 O.g.f. A(x) = (1 - *Sum_{n >= 1} binomial(2*n,n)*x^n)/(1 - 3/2*Sum_{n >= 1} binomial(2*n,n)*x^n). - Peter Bala, Sep 01 2016 MATHEMATICA CoefficientList[Series[(2-4*Sqrt[1-4*x])/(3-5*Sqrt[1-4*x]), {x, 0, 20}], x] (* Vaclav Kotesovec, Dec 09 2013 *) Flatten[{1, Table[FullSimplify[(2*n)!*Hypergeometric2F1Regularized[1, n+1/2, n+2, 16/25] / (25*n!) + 3*5^(2*n-1)/4^(n+1)], {n, 1, 20}]}] (* Vaclav Kotesovec, Dec 09 2013 *) CROSSREFS Cf. A000108, A001700, A049027, A076025. Sequence in context: A005668 A018904 A192807 * A161734 A081570 A122898 Adjacent sequences:  A076023 A076024 A076025 * A076027 A076028 A076029 KEYWORD nonn,easy AUTHOR N. J. A. Sloane, Oct 29 2002 STATUS approved

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Last modified January 17 19:58 EST 2019. Contains 319251 sequences. (Running on oeis4.)