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

0,2

Links

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

Vaclav Kotesovec, Recurrence

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

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 *)

Prog

(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))}

Crossrefs

Cf. A161797, A161798.

Sequence in context: A348200 A218092 A192479 * A378889 A378828 A182970

Adjacent sequences: A161796 A161797 A161798 * A161800 A161801 A161802

Keyword

nonn

Author

Paul D. Hanna, Jun 19 2009

Status

approved