login
The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

 

Logo

Thanks to everyone who made a donation during our annual appeal!
To see the list of donors, or make a donation, see the OEIS Foundation home page.

Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
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.

S. B. Ekhad, M. Yang, Proofs of Linear Recurrences of Coefficients of Certain Algebraic Formal Power Series Conjectured in the On-Line Encyclopedia Of Integer Sequences, (2017)

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

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

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