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 A078009 a(0)=1, for n>=1 a(n) = Sum_{k=0..n} 5^k*N(n,k) where N(n,k) = C(n,k)*C(n,k+1)/n are the Narayana numbers (A001263). 11
 1, 1, 6, 41, 306, 2426, 20076, 171481, 1500666, 13386206, 121267476, 1112674026, 10318939956, 96572168916, 910896992856, 8650566601401, 82644968321226, 793753763514806, 7659535707782916, 74225795172589006, 722042370787826076 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS More generally coefficients of (1 + m*x - sqrt(m^2*x^2 - (2*m+2)*x + 1) )/( 2*m*x ) are given by a(n) = Sum_{k=0..n} (m+1)^k * N(n,k). a(n) is the series reversion of x*(1-5*x)/(1-4*x). a(n+1) is the series reversion of x/(1 + 6*x + 5*x^2). a(n+1) counts (6,5)-Motzkin paths of length n, where there are 6 colors available for the H(1,0) steps and 5 for the U(1,1) steps. - Paul Barry, May 19 2005 The Hankel transform of this sequence is 5^C(n+1,2). - Philippe Deléham, Oct 29 2007 a(n) is the number of Schröder paths of semilength n in which there are no (2,0)-steps at level 0 and at a higher level they come in 4 colors. Example: a(2)=6 because we have UDUD, UUDD, UBD, UGD, URD, and UYD, where U=(1,1), D=(1,-1), while B, G, R, and Y are, respectively, blue, green, red, and yellow (2,0)-steps. - Emeric Deutsch, May 02 2011 Shifts left when INVERT transform applied five times. - Benedict W. J. Irwin, Feb 03 2016 LINKS Vincenzo Librandi, Table of n, a(n) for n = 0..200 Paul Barry, On Integer-Sequence-Based Constructions of Generalized Pascal Triangles, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.4. FORMULA G.f.: (1 + 4*x - sqrt(16*x^2 - 12*x + 1))/(10*x). a(n) = Sum_{k=0..n} A088617(n, k)*5^k*(-4)^(n-k). - Philippe Deléham, Jan 21 2004 With offset 1 : a(1)=1, a(n) = -4*a(n-1) + 5*Sum_{i=1..n-1} a(i)*a(n-i). - Benoit Cloitre, Mar 16 2004 a(n+1) = Sum_{k=0..floor(n/2)} C(n, 2*k)*C(k)*6^(n-2k)*5^k; - Paul Barry, May 19 2005 a(n) = ( 6*(2*n-1)*a(n-1) - 16*(n-2)*a(n-2) ) / (n+1) for n >= 2, a(0) = a(1) = 1. - Philippe Deléham, Aug 19 2005 From Gary W. Adamson, Jul 08 2011: (Start) a(n) = upper left term in M^n, M = the production matrix:   1, 1   5, 5, 5   1, 1, 1, 1   5, 5, 5, 5, 5   1, 1, 1, 1, 1, 1   ... (End) a(n) ~ sqrt(10+6*sqrt(5))*(6+2*sqrt(5))^n/(10*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Oct 13 2012 a(n) = A127848(n) for n > 0. - Philippe Deléham, Apr 03 2013 G.f.: 1/(1 - x/(1 - 5*x/(1 - x/(1 - 5*x/(1 - x/(1 - ...)))))), a continued fraction. - Ilya Gutkovskiy, Apr 21 2017 a(n) = hypergeom([1 - n, -n], [2], 5). - Peter Luschny, Mar 19 2018 MAPLE A078009_list := proc(n) local j, a, w; a := array(0..n); a[0] := 1; for w from 1 to n do a[w] := a[w-1]+5*add(a[j]*a[w-j-1], j=1..w-1) od; convert(a, list) end: A078009_list(20); # Peter Luschny, May 19 2011 MATHEMATICA Table[SeriesCoefficient[(1+4*x-Sqrt[16*x^2-12*x+1])/(10*x), {x, 0, n}], {n, 0, 30}] (* Vaclav Kotesovec, Oct 13 2012 *) a[n_] := Hypergeometric2F1[1 - n, -n, 2, 5]; Table[a[n], {n, 0, 30}] (* Peter Luschny, Mar 19 2018 *) PROG (PARI) a(n)=sum(k=0, n, 5^k/n*binomial(n, k)*binomial(n, k+1)) (MAGMA) R:=PowerSeriesRing(Rationals(), 30); Coefficients(R!( (1+4*x - Sqrt(16*x^2-12*x+1))/(10*x) )); // G. C. Greubel, Jun 28 2019 (MAGMA) [1] cat [&+[5^k*Binomial(n, k)*Binomial(n, k+1)/n:k in [0..n]]:n in [1..20]]; // Marius A. Burtea, Jan 21 2020 (Sage) a=((1+4*x -sqrt(16*x^2-12*x+1))/(10*x)).series(x, 30).coefficients(x, sparse=False); [1]+a[1:] # G. C. Greubel, Jun 28 2019 CROSSREFS Cf. A001003, A007564, A059231, A127848. Sequence in context: A186654 A152107 A143023 * A127848 A113573 A083161 Adjacent sequences:  A078006 A078007 A078008 * A078010 A078011 A078012 KEYWORD nonn AUTHOR Benoit Cloitre, May 10 2003 STATUS approved

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Last modified June 2 14:21 EDT 2020. Contains 334787 sequences. (Running on oeis4.)