A391463 Expansion of g/(2 - g^2)^2, where g = 1+x*g^3 is the g.f. of A001764.

1, 5, 33, 226, 1566, 10902, 76047, 530876, 3706626, 25876354, 180589843, 1259827146, 8784898992, 61229779780, 426567021204, 2970386408096, 20674948519258, 143842673407986, 1000344408813387, 6954033779333206, 48323287425710286, 335672254266040318

Offset

0,2

Links

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

Formula

a(n) = (1/(2*n)) * Sum_{k=1..n} k * (k+1) * Pell(k+2) * binomial(3*n,n-k) for n > 0.

D-finite with recurrence: (10368*n^2 + 10368*n + 2304)*a(n) + (71040*n^2 + 327936*n + 304128)*a(n + 1) + (-137760*n^2 - 632928*n - 699072)*a(n + 2) + (84048*n^2 + 500256*n + 724416)*a(n + 3) + (-25486*n^2 - 194278*n - 364188)*a(n + 4) + (4180*n^2 + 38916*n + 89432)*a(n + 5) + (-353*n^2 - 3859*n - 10416)*a(n + 6) + (12*n^2 + 150*n + 462)*a(n + 7) = 0. - Robert Israel, Dec 11 2025

Maple

f:= gfun:-rectoproc({(10368*n^2 + 10368*n + 2304)*a(n) + (71040*n^2 + 327936*n + 304128)*a(n + 1) + (-137760*n^2 - 632928*n - 699072)*a(n + 2) + (84048*n^2 + 500256*n + 724416)*a(n + 3) + (-25486*n^2 - 194278*n - 364188)*a(n + 4) + (4180*n^2 + 38916*n + 89432)*a(n + 5) + (-353*n^2 - 3859*n - 10416)*a(n + 6) + (12*n^2 + 150*n + 462)*a(n + 7), a(0) = 1, a(1) = 5, a(2) = 33, a(3) = 226, a(4) = 1566, a(5) = 10902, a(6) = 76047}, a(n), remember):
map(f, [$0..100]); # Robert Israel, Dec 11 2025

Mathematica

PellP[n_]:=Fibonacci[n, 2];
Table[If[n==0, 1, Sum[k *(k+1)*PellP[k+2]* Binomial[3 n, n-k], {k, 1, n}]/(2*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*(k+1)*pell(k+2)*binomial(3*n, n-k))/(2*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*(k+1)*(M^(k+2))[2, 1] * Binomial(3*n, n-k);
    end for;
    return S div (2*n);
end function;
seq := [ a(n) : n in [0..25] ];
seq; // Vincenzo Librandi, Dec 11 2025

Crossrefs

Cf. A391460, A391466.

Cf. A000129.

Sequence in context: A164538 A197533 A221441 * A083076 A361237 A162816

Adjacent sequences: A391460 A391461 A391462 * A391464 A391465 A391466

Keyword

nonn

Author

Seiichi Manyama, Dec 10 2025

Status

approved