A360103 a(n) = Sum_{k=0..n} binomial(n+4*k,n-k) * Catalan(k).

1, 2, 9, 49, 283, 1715, 10793, 69906, 463031, 3122264, 21363065, 147951489, 1035173405, 7306326465, 51959150713, 371950057003, 2678083379707, 19381867703946, 140915907625531, 1028760981192771, 7538511404971231, 55427326349613665, 408789584900354397

Offset

0,2

Links

Table of n, a(n) for n=0..22.

Formula

G.f. A(x) satisfies A(x) = 1/(1-x) + x * A(x)^2 / (1-x)^4.

G.f.: (1/(1-x)) * c(x/(1-x)^5), where c(x) is the g.f. of A000108.

D-finite with recurrence (n+1)*a(n) +2*(-5*n+3)*a(n-1) +(19*n-47)*a(n-2) +20*(-n+4)*a(n-3) +5*(3*n-17)*a(n-4) +2*(-3*n+22)*a(n-5) +(n-9)*a(n-6)=0. - R. J. Mathar, Mar 12 2023

Maple

A360103 := proc(n)
    add(binomial(n+4*k, n-k)*A000108(k), k=0..n) ;
end proc:
seq(A360103(n), n=0..40) ; # R. J. Mathar, Mar 12 2023

Prog

(PARI) a(n) = sum(k=0, n, binomial(n+4*k, n-k)*binomial(2*k, k)/(k+1));
(PARI) my(N=30, x='x+O('x^N)); Vec(2/((1-x)*(1+sqrt(1-4*x/(1-x)^5))))

Crossrefs

Partial sums of A360101.

Cf. A000108, A360057.

Cf. A006318, A007317, A162476, A360102.

Sequence in context: A295944 A356632 A224140 * A370345 A000167 A345750

Adjacent sequences: A360100 A360101 A360102 * A360104 A360105 A360106

Keyword

nonn

Author

Seiichi Manyama, Jan 25 2023

Status

approved