A360101 a(n) = Sum_{k=0..n} binomial(n+4*k-1,n-k) * Catalan(k).

1, 1, 7, 40, 234, 1432, 9078, 59113, 393125, 2659233, 18240801, 126588424, 887221916, 6271153060, 44652824248, 319990906290, 2306133322704, 16703784324239, 121534039921585, 887845073567240, 6509750423778460, 47888814944642434, 353362258550740732

Offset

0,3

Links

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

Formula

G.f. A(x) satisfies A(x) = 1 + x * A(x)^2 / (1-x)^5.

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

D-finite with recurrence (n+1)*a(n) +(-10*n+7)*a(n-1) +(19*n-56)*a(n-2) +10*(-2*n+9)*a(n-3) +5*(3*n-19)*a(n-4) +(-6*n+49)*a(n-5) +(n-10)*a(n-6)=0. - R. J. Mathar, Mar 12 2023

Maple

A360101 := proc(n)
    add(binomial(n+4*k-1, n-k)*A000108(k), k=0..n) ;
end proc:
seq(A360101(n), n=0..70) ; # R. J. Mathar, Mar 12 2023

Mathematica

m = 23;
A[_] = 0;
Do[A[x_] = 1 + x A[x]^2/(1 - x)^5 + O[x]^m // Normal, {m}];
CoefficientList[A[x], x] (* Jean-François Alcover, Aug 16 2023 *)

Prog

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

Crossrefs

Partial sums are A360103.

Cf. A000108, A360057.

Cf. A002212, A006319, A162475, A360100.

Sequence in context: A026782 A099459 A243875 * A371813 A051814 A154968

Adjacent sequences: A360098 A360099 A360100 * A360102 A360103 A360104

Keyword

nonn

Author

Seiichi Manyama, Jan 25 2023

Status

approved