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 A049027 G.f.: (1-2*x*C)/(1-3*x*C) where C = (1 - sqrt(1-4*x))/(2*x) is the g.f. for Catalan numbers A000108. 18
 1, 1, 4, 17, 74, 326, 1446, 6441, 28770, 128750, 576944, 2587850, 11615932, 52167688, 234383146, 1053386937, 4735393794, 21291593238, 95747347176, 430624242942, 1936925461644, 8712882517188, 39195738193836, 176335080590442 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS Row sums of triangle A035324. a(n+1) = {1,4,17,74,326, ...} is the binomial transform of A059738. - Philippe Deléham, Nov 26 2009 (1, 4, 17, 74, 326, ...) is the invert transform of the odd-indexed central binomial coefficients, A001700. - David Callan, Oct 14 2012 The sequence starting with index 1 is the INVERT transform of A001700: (1, 3, 10, 35, 126,...) and the second INVERT transform of the Catalan numbers starting with index 1: (1, 2, 5, 14, 42,...). - Gary W. Adamson, Jun 23 2015 REFERENCES L. W. Shapiro and C. J. Wang, Generating identities via 2 X 2 matrices, Congressus Numerantium, 205 (2010), 33-46. LINKS Vincenzo Librandi, Table of n, a(n) for n = 0..200 José Agapito, Ângela Mestre, Maria M. Torres, and Pasquale Petrullo, On One-Parameter Catalan Arrays, Journal of Integer Sequences, Vol. 18 (2015), Article 15.5.1 and arXiv:1505.05568 Paul Barry, Arnauld Mesinga Mwafise, Classical and Semi-Classical Orthogonal Polynomials Defined by Riordan Arrays, and Their Moment Sequences, Journal of Integer Sequences, Vol. 21 (2018), Article 18.1.5. Milan Janjić, Pascal Matrices and Restricted Words, J. Int. Seq., Vol. 21 (2018), Article 18.5.2. W. Lang, On generalizations of Stirling number triangles, J. Integer Seqs., Vol. 3 (2000), #00.2.4. J. W. Layman, The Hankel Transform and Some of its Properties, J. Integer Sequences, 4 (2001), #01.1.5. FORMULA G.f.: x*c(x)/(1-3*x*c(x)), c(x)= g.f. of Catalan numbers A000108. a(n+1) = Sum_{k=0..n} 2^k*comb(2n+1, n-k)*2*(k+1)/(n+k+2) - Paul Barry, Jun 22 2004 a(n) = (9*a(n-1) - Catalan(n-1))/2, n>1. - Vladeta Jovovic, Aug 08 2004 a(n+1) = Sum_{k=0..n} A039598(n,k)*2^k . - Philippe Deléham, Mar 21 2007 G.f.: 2 / (3 - 1 / sqrt(1 - 4*x)). - Michael Somos, Apr 08 2007 a(n) = Sum_{k=0..n} A039599(n,k)*A001045(k), for n>=1 . - Philippe Deléham, Jun 10 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,-3). [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: 4, 1, 0, 0, 0,... 1, 1, 1, 0, 0,... 1, 1, 1, 1, 0,... 1, 1, 1, 1, 1,... ... (End) Conjecture: 2*n*a(n) +(12-17*n)*a(n-1) +18*(2*n-3)*a(n-2)=0. - R. J. Mathar, Nov 14 2011 a(n) ~ 3^(2*n-1)/2^(n+1). - Vaclav Kotesovec, Oct 08 2012 0 = a(n)*(1296*a(n+1) - 1098*a(n+2) + 180*a(n+3)) + a(n+1)*(-126*a(n+1) + 253*a(n+2) - 58*a(n+3)) + a(n+2)*(-10*a(n+2) + 4*a(n+3)) if n>0. - Michael Somos, Jan 23 2014 O.g.f.: A(x) = 1/(1 - 1/2*Sum_{n >= 1} binomial(2*n,n)*x^n). - Peter Bala, Sep 01 2016 a(n) = 3^(2*n-1)/2^(n+1) + 2^n * (2*n-1)!! * hypergeom([1,n+1], [n+2], 8/9)/(9*(n+1)!) + 0^n * 2/3. - Vladimir Reshetnikov, Oct 08 2016 EXAMPLE G.f. = 1 + x + 4*x^2 + 17*x^3 + 74*x^4 + 326*x^5 + 1446*x^6 + 6441*x^7 + ... MATHEMATICA Table[SeriesCoefficient[2/(3-1/Sqrt[1-4*x]), {x, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Oct 08 2012 *) FunctionExpand@Table[3^(2n-1)/2^(n+1) + 2^n (2n-1)!! Hypergeometric2F1[1, n + 1/2, n + 2, 8/9]/(9 (n + 1)!) + 2 KroneckerDelta[n]/3, {n, 0, 20}] (* Vladimir Reshetnikov, Oct 08 2016 *) PROG (PARI) {a(n) = if( n<1, n==0, polcoeff( serreverse( x * (1 + 2*x) / (1 + 3*x)^2 + x * O(x^n) ), n))}; /* Michael Somos, Apr 08 2007 */ (PARI) {a(n) = if( n<0, 0, polcoeff( 2 / (3 - 1 / sqrt(1 - 4*x + x * O(x^n))), n))}; /* Michael Somos, Apr 08 2007 */ (MAGMA) [1] cat [n eq 1 select 1 else (9*Self(n-1)-Catalan(n-1))/2: n in [1..30]]; // Vincenzo Librandi, Jun 25 2015 (Sage) (2/(3-1/sqrt(1-4*x))).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, May 02 2019 CROSSREFS Cf. A000108, A001700. Sequence in context: A184700 A125586 A086351 * A026751 A227504 A218984 Adjacent sequences:  A049024 A049025 A049026 * A049028 A049029 A049030 KEYWORD easy,nonn AUTHOR STATUS approved

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Last modified October 20 15:15 EDT 2019. Contains 328267 sequences. (Running on oeis4.)