A161799 -id:A161799 - cp's OEIS frontend

A161798 G.f. satisfies: A(x) = 1/(1 - x/(1 - x*A(x))^3)^2.

1, 2, 9, 46, 262, 1590, 10081, 65986, 442518, 3024772, 20996141, 147603198, 1048747751, 7519252606, 54332565330, 395264527626, 2892666314150, 21281120904168, 157299607827727, 1167582500757800, 8699515577902203

Offset: 0

Paul D. Hanna, Jun 19 2009

nonn

G. C. Greubel, Table of n, a(n) for n = 0..1000
Vaclav Kotesovec, Recurrence

Cf. A161797, A161799.

Table[Sum[Binomial[2*n-k+1,k]/(n-k+1)*Binomial[n+2*k-1,n-k],{k,0,n}],{n,0,20}] (* Vaclav Kotesovec, Sep 18 2013 *)

{a(n,m=1)=sum(k=0,n,binomial(2*n-k+m,k)*m/(n-k+m)*binomial(n+2*k-1,n-k))}

a(n) = Sum_{k=0..n} C(2*n-k+1,k)/(n-k+1) * C(n+2*k-1,n-k).
Let A(x)^m = Sum_{n>=0} a(n,m)*x^n then
a(n,m) = Sum_{k=0..n} C(2*n-k+m,k)*m/(n-k+m) * C(n+2*k-1,n-k).
a(n) ~ c*d^n/(sqrt(Pi)*n^(3/2)), where d = 8.01957328653868383... is the root of the equation 3125 + 22356*d - 162432*d^2 - 361584*d^3 - 326592*d^4 + 46656*d^5 = 0 and c = 1.216730444416766043545857948227854793382399566... - Vaclav Kotesovec, Sep 18 2013

A349021 G.f. satisfies A(x) = 1/(1 - x/(1 - x*A(x))^2)^4.

Original entry on oeis.org

1, 4, 18, 104, 671, 4624, 33342, 248412, 1897219, 14774152, 116864936, 936390692, 7584216152, 61992689940, 510728310716, 4236545121924, 35354229533389, 296604036437692, 2500154435955614, 21164005790766980, 179841032283906149, 1533499916749203208

Offset: 0

Seiichi Manyama, Nov 06 2021

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..1000

Cf. A006013, A109081, A161799.

A349021 := proc(n)
    local s,t ;
    s := 2 ;
    t := 4;
    add( binomial(t*n-(t-1)*(k-1),k) * binomial(n+(s-1)*k-1,n-k) /(n-k+1) ,k=0..n) ;
end proc:
seq(A349021(n),n=0..40) ; # R. J. Mathar, May 12 2022

a(n, s=2, t=4) = sum(k=0, n, binomial(t*n-(t-1)*(k-1), k)*binomial(n+(s-1)*k-1, n-k)/(n-k+1));

If g.f. satisfies: A(x) = 1/(1 - x/(1 - x*A(x))^s)^t, then a(n) = Sum_{k=0..n} binomial(t*n-(t-1)*(k-1),k) * binomial(n+(s-1)*k-1,n-k)/(n-k+1).

cp's OEIS Frontend

A161798 G.f. satisfies: A(x) = 1/(1 - x/(1 - x*A(x))^3)^2.

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