A391464 Expansion of 1/(2 - g^2), where g = 1+x*g^4 is the g.f. of A002293.

1, 2, 13, 96, 753, 6120, 50927, 430992, 3694093, 31978904, 279059428, 2451295616, 21652137343, 192157404920, 1712319819482, 15312962637856, 137371506939205, 1235792077831416, 11144956230810508, 100736768308585216, 912394194390216484, 8279093240641977248

Offset

0,2

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Formula

a(n) = (1/n) * Sum_{k=1..n} k * Pell(k+1) * binomial(4*n,n-k) for n > 0.

Mathematica

PellP[n_]:=Fibonacci[n, 2];
Table[If[n==0, 1, Sum[k*PellP[k+1]*Binomial[4n, n-k], {k, 1, n}]/n], {n, 0, 20}] (* Vincenzo Librandi, Dec 11 2025 *)

Prog

(PARI) pell(n) = ([2, 1; 1, 0]^n)[2, 1];
a(n) = if(n==0, 1, sum(k=1, n, k*pell(k+1)*binomial(4*n, n-k))/n);
(Magma) M := Matrix(IntegerRing(), 2, 2, [2, 1, 1, 0]);
a := function(n)
    if n eq 0 then
        return 1;
    end if;
    S := 0;
    for k in [1..n] do
        S +:= k*(M^(k+1))[2, 1] * Binomial(4*n, n-k);
    end for;
    return S div (n);
end function;
seq := [ a(n) : n in [0..25] ];
seq; // Vincenzo Librandi, Dec 11 2025

Crossrefs

Cf. A289684, A391461.

Cf. A000129, A069271.

Sequence in context: A282724 A104255 A369202 * A118352 A389692 A372891

Adjacent sequences: A391461 A391462 A391463 * A391465 A391466 A391467

Keyword

nonn

Author

Seiichi Manyama, Dec 10 2025

Status

approved