A161799 G.f. satisfies: A(x) = 1/(1 - x/(1 - x*A(x))^2)^3.

1, 3, 12, 61, 345, 2085, 13182, 86106, 576543, 3936029, 27294390, 191722887, 1361291244, 9754412169, 70447946556, 512278417176, 3747570671685, 27561220671408, 203657352324178, 1511270129552163, 11257532921742528

Offset: 0

Author

Paul D. Hanna, Jun 19 2009

Keywords

• nonn

Links

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

Crossrefs

Cf. A161797, A161798.

Programs

Maple

A161799 := proc(n)
    local s,t ;
    s := 2 ;
    t := 3;
    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(A161799(n),n=0..40) ; # R. J. Mathar, May 12 2022

Mathematica

Table[Sum[Binomial[3*n-2*k+2,k]/(n-k+1)*Binomial[n+k-1,n-k],{k,0,n}],{n,0,20}] (* Vaclav Kotesovec, Sep 18 2013 *)

PARI

{a(n,m=1)=sum(k=0,n,binomial(3*n-2*k+3*m-1,k)*m/(n-k+m)*binomial(n+k-1,n-k))}

Formula

a(n) = Sum_{k=0..n} C(3*n-2*k+2,k)/(n-k+1) * C(n+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(3*n-2*k+3*m-1,k)*m/(n-k+m) * C(n+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 = 1.56703431595354192843152170651865561188... - Vaclav Kotesovec, Sep 18 2013