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A098616 Product of Pell and Catalan numbers: a(n) = A000129(n+1)*A000108(n). 12
1, 2, 10, 60, 406, 2940, 22308, 175032, 1408550, 11561836, 96425836, 814773960, 6960289532, 60012947800, 521582661000, 4564643261040, 40190674554630, 355772529165900, 3164408450118300, 28266363849505320, 253466716153665300 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

Radius of convergence: r = (sqrt(2)-1)/4, where A(r) = sqrt(2+sqrt(2)).

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

Harvey P. Dale, Table of n, a(n) for n = 0..1000

Index entries for linear recurrences with constant coefficients, signature (2, 1).

FORMULA

G.f.: A(x) = sqrt( (1-4*x - sqrt(1-8*x-16*x^2))/16 )/x.

Run lengths of zeros (mod 10) equal (5^k - (-1)^k)/2 - 1 starting at index (5^k + (-1)^k)/2:

a(n) == 0 (mod 10) for n = (5^k + (-1)^k)/2 through n = 5^k - 1 when k>=1.

a(n) ~ 2^(2*n-3/2) * (1+sqrt(2))^(n+1) / (sqrt(Pi) * n^(3/2)). - Vaclav Kotesovec, May 09 2014

A(-x) = 1/x * series reversion( x*(2*x + sqrt(1 - 4*x^2)) ). Compare with the o.g.f. B(x) of the central binomial numbers A000984, which satisfies B(-x) = 1/x * series reversion( x*(2*x + sqrt(1 + 4*x^2)) ). See also A214377. - Peter Bala, Oct 19 2015

EXAMPLE

Sequence begins: [1*1, 2*1, 5*2, 12*5, 29*14, 70*42, 169*132, 408*429,...].

MATHEMATICA

With[{nn=30}, Times@@@Thread[{LinearRecurrence[{2, 1}, {1, 2}, nn], CatalanNumber[ Range[0, nn-1]]}]] (* Harvey P. Dale, Jan 04 2012 *)

PROG

(PARI) {a(n)=binomial(2*n, n)/(n+1)*round(((1+sqrt(2))^(n+1)-(1-sqrt(2))^(n+1))/(2*sqrt(2)))}

CROSSREFS

Cf. A000129, A000108, A098614, A098617, A098618, A000984, A214377.

Sequence in context: A301625 A262001 A276310 * A082042 A260657 A079856

Adjacent sequences:  A098613 A098614 A098615 * A098617 A098618 A098619

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Oct 09 2004

STATUS

approved

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Last modified October 22 15:45 EDT 2018. Contains 316490 sequences. (Running on oeis4.)