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A216541 Product of Lucas and Catalan numbers: a(n) = A000032(n+1)*A000108(n). 1
1, 3, 8, 35, 154, 756, 3828, 20163, 108680, 598026, 3342404, 18929092, 108374252, 626264700, 3647936160, 21396522915, 126262239570, 749087596620, 4465444206300, 26733390275130, 160663411399920, 968937572793060, 5862111195487560, 35569106862459300, 216395609659221564 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
LINKS
FORMULA
G.f.: (1 - sqrt( (1-2*x + sqrt(1-4*x-16*x^2))/2 )) / x.
G.f. satisfies: A(x) = (2+5*x) - (1+4*x)*A(x) + x*(5+2*x)*A(x)^2 - 4*x^2*A(x)^3 + x^3*A(x)^4.
n*(n+1)*a(n) -2*n*(2n-1)*a(n-1) -4*(2*n-1)*(2*n-3)*a(n-2)=0. - R. J. Mathar, Sep 11 2012
Sum_{n>=0} a(n)/8^n = 8 - 2*sqrt(10). - Amiram Eldar, May 05 2023
EXAMPLE
G.f.: A(x) = 1 + 3*x + 8*x^2 + 35*x^3 + 154*x^4 + 756*x^5 + 3828*x^6 +...
such that the coefficients equal the term-wise products:
A = [1*1, 3*1, 4*2, 7*5, 11*14, 18*42, 29*132, 47*429, 76*1430, ...].
MATHEMATICA
a[n_] := LucasL[n+1] * CatalanNumber[n]; Array[a, 25, 0] (* Amiram Eldar, May 05 2023 *)
PROG
(PARI) {a(n)=(2*fibonacci(n)+fibonacci(n+1))*binomial(2*n, n)/(n+1)}
(PARI) {a(n)=polcoeff( (1 - sqrt( (1-2*x + sqrt(1-4*x-16*x^2 +x^2*O(x^n)))/2 )) / x, n)}
for(n=0, 25, print1(a(n), ", "))
CROSSREFS
Sequence in context: A148918 A304726 A226679 * A347896 A194090 A294385
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Sep 08 2012
STATUS
approved

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Last modified July 19 19:23 EDT 2024. Contains 374433 sequences. (Running on oeis4.)