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 A178792 Dot product of the rows of triangle A046899 with vector (1,2,4,8,...) (= A000079). 5
 1, 5, 31, 209, 1471, 10625, 78079, 580865, 4361215, 32978945, 250806271, 1916280833, 14698053631, 113104519169, 872801042431, 6751535300609, 52337071357951, 406468580343809, 3162019821780991, 24634626678980609, 192179216026959871 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS Hankel transform is A133460. LINKS Vincenzo Librandi, Table of n, a(n) for n = 0..200 J. Abate and W. Whitt, Brownian motion and generalized Catalan numbers, Journal of Integer Sequences, Vol. 14 (2011). Karl Dilcher, Maciej Ulas, Arithmetic properties of polynomial solutions of the Diophantine equation P(x)x^(n+1)+Q(x)(x+1)^(n+1) = 1, arXiv:1909.11222 [math.NT], 2019. FORMULA a(n) = Sum_{k = 0..n} A046899(n,k)*2^k = Sum_{k = 0..n} 2^k*binomial(n+k,k). G.f.: (1/3)*(4/sqrt(1 - 8*x) - 1/(1 - x*c(2*x))) with c(x) the g.f. of the Catalan numbers A000108. a(n) = (1/3)*(4*2^n*A000984(n) - A064062(n)). a(n) + a(n+1) = 6*2^n*A001700(n). O.g.f.: (3 - sqrt(1 - 8*x))/(2*(1 + x)*sqrt(1 - 8*x)). - Peter Bala, Apr 10 2012 a(n) = (2^n)*binomial(2+2*n,1+n)*2F1(1, 2+2*n; 2+n)(-1). - Olivier Gérard, Aug 19 2012 n*a(n) = (7*n - 4)*a(n-1) + 4*(2*n - 1)*a(n-2). - Vaclav Kotesovec, Oct 20 2012 a(n) ~ 2^(3n+2)/(3*sqrt(Pi*n)). - Vaclav Kotesovec, Oct 20 2012 a(n) = Sum_{k = 0..n} binomial(k+n,n)*binomial(2*n+1,n-k). - Vladimir Kruchinin, Oct 28 2016 a(n) = 1/2*(n + 1)*binomial(2*n+2,n+1)*Sum_{k = 0..n} binomial(n,k)/(n + k + 1). - Peter Bala, Feb 21 2017 a(n) = binomial(2*n+2,n+1)*hypergeom([-n, n+1], [n+2], -1)/2. - Peter Luschny, Feb 21 2017 a(n) = (-1)^(n+1) - 2^(n+1)*(2*n+1)*binomial(2*n,n)*hypergeom([1, 2*n+2], [n+2], 2)/(n+1). - John M. Campbell, Jul 14 2018 EXAMPLE a(3) = (1,4,10,20)dot(1,2,4,8) = 209. MAPLE a := n -> binomial(2*n+2, n+1)*hypergeom([-n, n + 1], [n + 2], -1)/2: seq(simplify(a(n)), n=0..20); # Peter Luschny, Feb 21 2017 MATHEMATICA CoefficientList[Series[(3-Sqrt[1-8*x])/(2*(1+x)*Sqrt[1-8*x]), {x, 0, 20}], x] (* Vaclav Kotesovec, Oct 20 2012 *) Table[Sum[2^k*Binomial[n + k, k], {k, 0, n}], {n, 0, 20}] (* Michael De Vlieger, Oct 28 2016 *) a[n_] := (-1)^(n + 1) - 2^(n + 1) (2n + 1) Binomial[2n, n] Hypergeometric2F1[1, 2n + 2, n + 2, 2]/(n + 1); Array[a, 22, 0] (* Robert G. Wilson v, Jul 21 2018 *) CROSSREFS Row sums of A091811. Sequence in context: A296032 A301420 A092636 * A007197 A247639 A002649 Adjacent sequences:  A178789 A178790 A178791 * A178793 A178794 A178795 KEYWORD nonn,easy AUTHOR Joseph Abate, Jun 15 2010 STATUS approved

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Last modified August 12 08:33 EDT 2020. Contains 336438 sequences. (Running on oeis4.)