A365128 G.f. satisfies A(x) = (1 + x*A(x)*(1 + x*A(x)))^3.

1, 3, 15, 88, 567, 3876, 27607, 202653, 1522365, 11647038, 90435804, 710855544, 5645365576, 45228648078, 365109237801, 2966862631856, 24248879149005, 199213507774365, 1644138419038500, 13625326165675698, 113336685917785332, 945931091151789808

Offset

0,2

Links

Table of n, a(n) for n=0..21.

Formula

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

D-finite with recurrence 205*(5*n+6)*(5*n+2)*(5*n+3)*(5*n+4)*(n+1)*a(n) +9*(-5948592*n^5+11369145*n^4 -5182620*n^3 -351495*n^2+204302*n-6560) *a(n-1) +243*(-801282*n^5 +14391105*n^4 -55889790*n^3 +90254895*n^2 -66199848*n +18182560)*a(n-2) +6561*(3*n-5) *(3*n-4)*(93048*n^3 -579621*n^2 +1227037*n -878874)*a(n-3) +48715425*(n-3) *(3*n-4)*(3*n-7) *(3*n-5)*(3*n-8)*a(n-4)=0. - R. J. Mathar, Dec 04 2023

From Seiichi Manyama, Sep 20 2024: (Start)

G.f.: (1/x) * Series_Reversion( x / (1+x+x^2)^3 ).

G.f.: B(x)^3, where B(x) is the g.f. of A255673. (End)

Maple

A365128 := proc(n)
    add(binomial(3*(n+1), k) * binomial(k, n-k), k=0..n) ;
    %/(n+1) ;
end proc:
seq(A365128(n), n=0..70) ; # R. J. Mathar, Dec 04 2023

Prog

(PARI) a(n, s=1, t=3) = sum(k=0, n, binomial(t*(n+1), k)*binomial(s*k, n-k))/(n+1);

Crossrefs

Cf. A001006, A143927.

Cf. A019497, A255673.

Sequence in context: A192253 A368972 A370472 * A127785 A287511 A231257

Adjacent sequences: A365125 A365126 A365127 * A365129 A365130 A365131

Keyword

nonn

Author

Seiichi Manyama, Aug 23 2023

Status

approved