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 A082302 Expansion of g.f.: (1 - 5*x - sqrt(25*x^2 - 14*x + 1))/(2*x). 6
 1, 6, 42, 330, 2814, 25422, 239442, 2326434, 23151030, 234784662, 2417832186, 25216231866, 265796560302, 2827138163550, 30306009654690, 327081253546770, 3551148743559270, 38758882760119590, 425024567305557450 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS More generally coefficients of (1 - m*x - sqrt(m^2*x^2 - (2*m + 4)*x + 1))/(2*x) are given by a(0)=1 and, for n > 0, a(n) = (1/n)*Sum_{k=0..n} (m+1)^k*C(n,k)*C(n,k-1). Hankel transform is 6^C(n+1,2). - Philippe Deléham, Feb 11 2009 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. Paul Barry, Generalized Catalan Numbers Associated with a Family of Pascal-like Triangles, J. Int. Seq., Vol. 22 (2019), Article 19.5.8. FORMULA Equals 6*A078018(n) for n > 0. a(0)=1; for n > 0, a(n) = (1/n)*Sum_{k=0..n} 6^k*C(n, k)*C(n, k-1). D-finite with recurrence: (n+1)*a(n) + 7*(1-2n)*a(n-1) + 25*(n-2)*a(n-2) = 0. - R. J. Mathar, Dec 08 2011 a(n) ~ sqrt(12 + 7*sqrt(6))*(7 + 2*sqrt(6))^n/(2*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Oct 14 2012 a(n) = 6*hypergeom([1 - n, -n], [2], 6) for n > 0. - Peter Luschny, May 22 2017 G.f.: 1/(1 - 5*x - x/(1 - 5*x - x/(1 - 5*x - x/(1 - 5*x - x/(1 - ...))))), a continued fraction. - Ilya Gutkovskiy, Apr 04 2018 MAPLE A082302_list := proc(n) local j, a, w; a := array(0..n); a[0] := 1; for w from 1 to n do a[w] := 6*a[w-1]+add(a[j]*a[w-j-1], j=1..w-1) od; convert(a, list)end: A082302_list(18); # Peter Luschny, May 19 2011 a := n -> `if`(n=0, 1, 6*hypergeom([1 - n, -n], [2], 6)): seq(simplify(a(n)), n=0..18); # Peter Luschny, May 22 2017 MATHEMATICA Table[SeriesCoefficient[(1-5*x-Sqrt[25*x^2-14*x+1])/(2*x), {x, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Oct 14 2012 *) PROG (PARI) a(n)=if(n<1, 1, sum(k=0, n, 6^k*binomial(n, k)*binomial(n, k-1))/n) (PARI) x='x+O('x^99); Vec((1-5*x-(25*x^2-14*x+1)^(1/2))/(2*x)) \\ Altug Alkan, Apr 04 2018 (GAP) Concatenation([1], List([1..20], n->(1/n)*Sum([0..n], k->6^k*Binomial(n, k)*Binomial(n, k-1)))); # Muniru A Asiru, Apr 05 2018 (MAGMA) m:=50; R:=PowerSeriesRing(Rationals(), m); Coefficients(R!((1-5*x-Sqrt(25*x^2-14*x+1))/(2*x))); // G. C. Greubel, Aug 16 2018 CROSSREFS Cf. A006318, A047891. Sequence in context: A033296 A218755 A165314 * A144223 A320758 A262671 Adjacent sequences:  A082299 A082300 A082301 * A082303 A082304 A082305 KEYWORD nonn AUTHOR Benoit Cloitre, May 10 2003 STATUS approved

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Last modified August 9 01:38 EDT 2022. Contains 356016 sequences. (Running on oeis4.)