A185020 a(n) = A000108(n)*A002605(n+1), where A000108 are the Catalan numbers.

1, 2, 12, 80, 616, 5040, 43296, 384384, 3500640, 32517056, 306896512, 2934597120, 28369508608, 276810483200, 2722537128960, 26963147796480, 268659456837120, 2691301381401600, 27089160416102400, 273833161582632960, 2778754123765002240, 28296326851107594240

Offset

0,2

Comments

More generally, given {S} such that: S(n) = b*S(n-1) + c*S(n-2), |b|>0, |c|>0, S(0)=1, 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.

Conjecture: +n*(n+1)*a(n) -4*n*(2*n-1)*a(n-1) -8*(2*n-1)*(2*n-3)*a(n-2)=0. - R. J. Mathar, Oct 08 2016

Links

G. C. Greubel, Table of n, a(n) for n = 0..960

Formula

G.f.: sqrt( (1-4*x - sqrt(1-8*x-32*x^2))/24 )/x.

Example

G.f.: A(x) = 1 + 1*2*x + 2*6*x^2 + 5*16*x^3 + 14*44*x^4 + 42*120*x^5 + 132*328*x^6 +...+ A000108(n)*A002605(n+1)*x^n +...

Mathematica

MapIndexed[CatalanNumber[#2 - 1] #1 &, Rest@ RecurrenceTable[{a[n] == 2 (a[n - 1] + a[n - 2]), a[0] == 0, a[1] == 1}, a, {n, 22}]] // Flatten (* or *)
CoefficientList[Series[Sqrt[(1 - 4 x - Sqrt[1 - 8 x - 32 x^2])/24]/x, {x, 0, 21}], x] (* Michael De Vlieger, Oct 08 2016 *)

Prog

(PARI) {A000108(n)=binomial(2*n, n)/(n+1)}
{A002605(n)=polcoeff(x/(1-2*x-2*x^2 +x*O(x^n)), n)}
{a(n)=A000108(n)*A002605(n+1)}
for(n=0, 30, print1(a(n), ", "))

Crossrefs

Cf. A002605.

Sequence in context: A069723 A063481 A274782 * A052822 A246018 A292933

Adjacent sequences: A185017 A185018 A185019 * A185021 A185022 A185023

Keyword

nonn

Author

Paul D. Hanna, Dec 26 2012

Status

approved