A364336 G.f. satisfies A(x) = (1 + x) * (1 + x*A(x)^3).

1, 2, 7, 39, 242, 1634, 11631, 85957, 653245, 5072862, 40077807, 321106623, 2602911282, 21308131235, 175909559897, 1462846379247, 12242600576066, 103035285071630, 871490142773640, 7404121610615520, 63157400073057627, 540689217572662413, 4644083121177225292

Offset: 0

Author

Seiichi Manyama, Jul 19 2023

Keywords

• nonn

Links

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

Crossrefs

Cf. A073157, A364337, A364338, A364339.

Cf. A198953, A212071, A215623, A215654, A216359.

Programs

Maple

A364336 := proc(n)
    add( binomial(3*k+1,k) * binomial(3*k+1,n-k)/(3*k+1),k=0..n) ;
end proc:
seq(A364336(n),n=0..80); # R. J. Mathar, Jul 25 2023

Mathematica

nmax = 80; A[_] = 1;
Do[A[x_] = (1 + x)*(1 + x*A[x]^3) + O[x]^(nmax+1) // Normal, {nmax+1}];
CoefficientList[A[x], x] (* Jean-François Alcover, Mar 03 2024 *)

PARI

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

Formula

a(n) = Sum_{k=0..n} binomial(3*k+1,k) * binomial(3*k+1,n-k) / (3*k+1).

D-finite with recurrence -2*n*(2*n+1)*a(n) +(3*n^2+23*n-14)*a(n-1) +(207*n^2 -635*n +494)*a(n-2) +2*(397*n^2 -2031*n +2600)*a(n-3) +6*(75*n-244) *(3*n-11)*a(n-4) +9*(45*n-179) *(3*n-14)*a(n-5) +63*(3*n-14) *(3*n-17)*a(n-6) +12*(3*n-16) *(3*n-20)*a(n-7)=0. - R. J. Mathar, Jul 25 2023

From Peter Bala, Sep 10 2024: (Start)

x/series_reversion(x*A(x)) = 1 + 2*x + 3*x^2 + 13*x^3 + 32*x^4 + 147*x^5 + ..., the g.f. of A216359.

(1/x) * series_reversion(x/A(x)) = 1 + 2*x + 11*x^2 + 89*x^3 + 836*x^4 + 8551*x^5 + ..., the g.f. of A215623. (End)