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A360045
a(n) = Sum_{k=0..floor(n/3)} binomial(n+2,3*k+2) * Catalan(k).
4
1, 3, 6, 11, 21, 42, 86, 180, 387, 852, 1908, 4329, 9933, 23013, 53748, 126386, 298953, 710847, 1698086, 4073181, 9806565, 23689555, 57401322, 139475190, 339767545, 829638396, 2030206248, 4978136133, 12229451631, 30095772966, 74184390468, 183139941384
OFFSET
0,2
FORMULA
a(n) = binomial(n+2,2) + Sum_{k=0..n-3} a(k) * a(n-k-3).
G.f. A(x) satisfies: A(x) = 1/(1-x)^3 + x^3 * A(x)^2.
G.f.: 2 / ( (1-x) * ((1-x)^2 + sqrt((1-x)^4 - 4*x^3*(1-x))) ).
D-finite with recurrence (n+3)*a(n) +4*(-n-2)*a(n-1) +6*(n+1)*a(n-2) +2*(-4*n+3)*a(n-3) +5*(n-1)*a(n-4)=0. - R. J. Mathar, Jan 25 2023
PROG
(PARI) a(n) = sum(k=0, n\3, binomial(n+2, 3*k+2)*binomial(2*k, k)/(k+1));
(PARI) my(N=40, x='x+O('x^N)); Vec(2/((1-x)*((1-x)^2+sqrt((1-x)^4-4*x^3*(1-x)))))
CROSSREFS
KEYWORD
nonn
AUTHOR
Seiichi Manyama, Jan 23 2023
STATUS
approved