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 A200375 Product of Catalan and Jacobsthal numbers: a(n) = A000108(n)*A001045(n). 4
 1, 1, 6, 25, 154, 882, 5676, 36465, 244530, 1657942, 11471668, 80242890, 568080772, 4056976900, 29212908120, 211783889025, 1544811959970, 11328491394990, 83473572128100, 617702666484750, 4588654943721420, 34206312386929020, 255803818897858920, 1918528298674328250, 14427334095935095764 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS More generally, given {S} such that: S(n) = b*S(n-1) + c*S(n-2), |b|>0, |c|>0, then Sum_{n>=0} S(n)*Catalan(n)*x^n = sqrt( (1-2*b*x - sqrt(1-4*b*x-16*c*x^2))/(2*b^2+8*c) )/x. LINKS Michael De Vlieger, Table of n, a(n) for n = 0..1112 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. FORMULA G.f.: sqrt( (1-2*x - sqrt(1-4*x-32*x^2))/2 )/(3*x). G.f.: (1/x)*Series_Reversion(x-x^2 - 4*x^3*Sum_{n>=0} A000108(n)*3^n*x^(2*n) ). G.f. satisfies: A(x) = G(x*A(x)) and G(x) = A(x/G(x)) where G(x) is the g.f. of A200376: G(x) = 1/sqrt(1-10*x^2 + x^4/(1-8*x^2)) + x/(1-9*x^2). n*(n+1)*a(n) -2*n*(2*n-1)*a(n-1) -8*(2*n-1)*(2*n-3)*a(n-2)=0. - R. J. Mathar, Nov 17 2011 a(n) = binomial(2*n,n)/(n+1) * (2^(n+1) + (-1)^n)/3. EXAMPLE G.f.: A(x) = 1 + x + 2*3*x^2 + 5*5*x^3 + 14*11*x^4 + 42*21*x^5 + 132*43*x^6 + 429*85*x^7 + 1430*171*x^8 +...+ A000108(n)*A001045(n)*x^n +... The g.f. of the Jacobsthal sequence A001045, F(x) = 1/(1-x-2*x^2), begins: F(x) = 1 + x + 3*x^2 + 5*x^3 + 11*x^4 + 21*x^5 + 43*x^6 + 85*x^7 + 171*x^8 +... The g.f. of A200376, where G(x) =  A(x/G(x)), begins: G(x) = 1 + x + 5*x^2 + 9*x^3 + 37*x^4 + 81*x^5 + 301*x^6 + 729*x^7 +... in which the odd-indexed coefficients are powers of 9. MATHEMATICA Array[CatalanNumber[# - 1] (2^# - (-1)^#)/3 &, 25] (* Michael De Vlieger, Apr 24 2018 *) PROG (PARI) {a(n)=binomial(2*n, n)/(n+1)*(2^(n+1)+(-1)^n)/3} (PARI) {a(n)=polcoeff(sqrt((1-2*x - sqrt(1-4*x-32*x^2+O(x^(n+3))))/2)/(3*x), n)} (PARI) {a(n)=polcoeff((1/x)*serreverse(x-x^2 - 4*x^3*sum(m=0, n\2, binomial(2*m, m)/(m+1)*3^m*x^(2*m))+x^3*O(x^n)), n)} CROSSREFS Cf. A200376, A098614, A098616, A200312. Sequence in context: A012293 A012594 A242858 * A009464 A042529 A323132 Adjacent sequences:  A200372 A200373 A200374 * A200376 A200377 A200378 KEYWORD nonn AUTHOR Paul D. Hanna, Nov 16 2011 STATUS approved

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Last modified January 18 16:27 EST 2020. Contains 331011 sequences. (Running on oeis4.)