A200375 Product of Catalan and Jacobsthal numbers: a(n) = A000108(n)*A001045(n+1).

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

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, On the duals of the Fibonacci and Catalan-Fibonacci polynomials and Motzkin paths, arXiv:2101.10218 [math.CO], 2021.

Paul Barry and 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 and 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.

From Peter Bala, Aug 17 2021: (Start)

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

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.

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.

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

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: A012594 A357089 A242858 * A009464 A042529 A323132

Adjacent sequences: A200372 A200373 A200374 * A200376 A200377 A200378

Keyword

nonn

Author

Paul D. Hanna, Nov 16 2011

Extensions

Typo in Name corrected by Peter Bala, Aug 17 2021

Status

approved