A391463 - OEIS

%I #26 Dec 17 2025 17:26:26

%S 1,5,33,226,1566,10902,76047,530876,3706626,25876354,180589843,

%T 1259827146,8784898992,61229779780,426567021204,2970386408096,

%U 20674948519258,143842673407986,1000344408813387,6954033779333206,48323287425710286,335672254266040318

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

%H Vincenzo Librandi, <a href="/A391463/b391463.txt">Table of n, a(n) for n = 0..1000</a>

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

%F 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

%p 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):

%p map(f, [$0..100]); # _Robert Israel_, Dec 11 2025

%t PellP[n_]:=Fibonacci[n, 2];

%t 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 *)

%o (PARI) pell(n) = ([2, 1; 1, 0]^n)[2, 1];

%o a(n) = if(n==0, 1, sum(k=1, n, k*(k+1)*pell(k+2)*binomial(3*n, n-k))/(2*n));

%o (Magma) M := Matrix(IntegerRing(), 2, 2, [2,1,1,0]);

%o a := function(n)

%o     if n eq 0 then

%o         return 1;

%o     end if;

%o     S := 0;

%o     for k in [1..n] do

%o         S +:= k*(k+1)*(M^(k+2))[2,1] * Binomial(3*n, n-k);

%o     end for;

%o     return S div (2*n);

%o end function;

%o seq := [ a(n) : n in [0..25] ];

%o seq; // _Vincenzo Librandi_, Dec 11 2025

%Y Cf. A391460, A391466.

%Y Cf. A000129.

%K nonn

%O 0,2

%A _Seiichi Manyama_, Dec 10 2025