A200375 - OEIS

%I #39 Dec 14 2024 07:08:47

%S 1,1,6,25,154,882,5676,36465,244530,1657942,11471668,80242890,

%T 568080772,4056976900,29212908120,211783889025,1544811959970,

%U 11328491394990,83473572128100,617702666484750,4588654943721420,34206312386929020,255803818897858920,1918528298674328250,14427334095935095764

%N Product of Catalan and Jacobsthal numbers: a(n) = A000108(n)*A001045(n+1).

%C 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.

%H Michael De Vlieger, <a href="/A200375/b200375.txt">Table of n, a(n) for n = 0..1112</a>

%H Paul Barry, <a href="https://arxiv.org/abs/2101.10218">On the duals of the Fibonacci and Catalan-Fibonacci polynomials and Motzkin paths</a>, arXiv:2101.10218 [math.CO], 2021.

%H Paul Barry and Arnauld Mesinga Mwafise, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL21/Barry/barry362.html">Classical and Semi-Classical Orthogonal Polynomials Defined by Riordan Arrays, and Their Moment Sequences</a>, Journal of Integer Sequences, Vol. 21 (2018), Article 18.1.5.

%H S. B. Ekhad and M. Yang, <a href="http://sites.math.rutgers.edu/~zeilberg/tokhniot/oMathar1maple12.txt">Proofs of Linear Recurrences of Coefficients of Certain Algebraic Formal Power Series Conjectured in the On-Line Encyclopedia Of Integer Sequences</a>, (2017).

%F G.f.: sqrt( (1-2*x - sqrt(1-4*x-32*x^2))/2 )/(3*x).

%F G.f.: (1/x)*Series_Reversion(x-x^2 - 4*x^3*Sum_{n>=0} A000108(n)*3^n*x^(2*n) ).

%F 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).

%F 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

%F a(n) = binomial(2*n,n)/(n+1) * (2^(n+1) + (-1)^n)/3.

%F From _Peter Bala_, Aug 17 2021: (Start)

%F G.f.: A(x) = (sqrt(1 + 4*x) - sqrt(1 - 8*x))/(6*x).

%F A(x) = 1/sqrt(1 + 4*x)*c( 3*x/(1 + 4*x) ), where c(x) = (1 - sqrt(1- 4*x))/(2*x) is the g.f. of the Catalan numbers A000108. Cf. A151374.

%F In general, [x^n] ( 1/sqrt(1 + 4*x)*c( k*x/(1 + 4*x) ) ) = Catalan(n)*((k-1)^(n+1) + (-1)^(n+1))/k.

%F A(x) = 1/sqrt(1 - 8*x)*c( -3*x/(1 - 8*x) ). (End)

%F G.f. A(x) satisfies A(x) = sqrt( 1 + 2*x*A(x)^2 + 9*x^2*A(x)^4 ). - _Paul D. Hanna_, Dec 14 2024

%e 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 +...

%e The g.f. of the Jacobsthal sequence A001045, F(x) = 1/(1-x-2*x^2), begins:

%e 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 +...

%e The g.f. of A200376, where G(x) =  A(x/G(x)), begins:

%e G(x) = 1 + x + 5*x^2 + 9*x^3 + 37*x^4 + 81*x^5 + 301*x^6 + 729*x^7 +...

%e in which the odd-indexed coefficients are powers of 9.

%t Array[CatalanNumber[# - 1] (2^# - (-1)^#)/3 &, 25] (* _Michael De Vlieger_, Apr 24 2018 *)

%o (PARI) {a(n) = binomial(2*n, n)/(n+1) * (2^(n+1) + (-1)^n)/3}

%o (PARI) {a(n) = polcoef(sqrt((1-2*x - sqrt(1-4*x-32*x^2 +O(x^(n+3))))/2)/(3*x), n)}

%o (PARI) {a(n) = polcoef((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)}

%Y Cf. A200376, A098614, A098616, A200312.

%K nonn

%O 0,3

%A _Paul D. Hanna_, Nov 16 2011

%E Typo in Name corrected by _Peter Bala_, Aug 17 2021