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A360059
a(n) = Sum_{k=0..n} (-1)^k * binomial(n+3*k+3,n-k) * Catalan(k).
2
1, 3, 4, 3, 5, 12, 6, -13, 29, 95, -130, -304, 895, 1050, -5068, -2181, 27743, -5481, -143532, 117983, 700831, -1074414, -3163138, 7872784, 12585117, -51587107, -38040886, 312988334, 18178883, -1779688404, 1013771196, 9485832411, -11749675733, -46878057651
OFFSET
0,2
FORMULA
a(n) = binomial(n+3,3) - Sum_{k=0..n-1} a(k) * a(n-k-1).
G.f. A(x) satisfies A(x) = 1/(1-x)^4 - x * A(x)^2.
G.f.: 2 / ( (1-x)^4 * (1 + sqrt( 1 + 4*x/(1-x)^4 )) ).
D-finite with recurrence (n+1)*a(n) +(-n-2)*a(n-1) +6*(n-2)*a(n-2) +10*(-n+2)*a(n-3) +5*(n-3)*a(n-4) +(-n+4)*a(n-5)=0. - R. J. Mathar, Jan 25 2023
MATHEMATICA
Table[Sum[(-1)^k Binomial[n+3k+3, n-k]CatalanNumber[k], {k, 0, n}], {n, 0, 40}] (* Harvey P. Dale, May 06 2024 *)
PROG
(PARI) a(n) = sum(k=0, n, (-1)^k*binomial(n+3*k+3, n-k)*binomial(2*k, k)/(k+1));
(PARI) my(N=40, x='x+O('x^N)); Vec(2/((1-x)^4*(1+sqrt(1+4*x/(1-x)^4))))
CROSSREFS
KEYWORD
sign
AUTHOR
Seiichi Manyama, Jan 23 2023
STATUS
approved