A391462 - OEIS

%I #16 Feb 12 2026 19:59:21

%S 1,3,16,95,590,3755,24266,158448,1042346,6895491,45814172,305442703,

%T 2042080348,13684126812,91875463080,617863215368,4160959900522,

%U 28055667406147,189367714620976,1279362095322717,8650402250562118,58532189534911523,396310088499317090

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

%H Robert Israel, <a href="/A391462/b391462.txt">Table of n, a(n) for n = 0..1193</a>

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

%F D-finite with recurrence: (5184*n^2 + 5184*n + 1152)*a(n) + (4416*n^2 + 34368*n + 36288)*a(n + 1) + (-13728*n^2 - 70032*n - 85824)*a(n + 2) + (6672*n^2 + 42312*n + 66936)*a(n + 3) + (-1325*n^2 - 10499*n - 20850)*a(n + 4) + (119*n^2 + 1148*n + 2772)*a(n + 5) + (-4*n^2 - 46*n - 132)*a(n + 6) = 0. - _Robert Israel_, Feb 12 2026

%p f:= gfun:-rectoproc({(5184*n^2 + 5184*n + 1152)*a(n) + (4416*n^2 + 34368*n + 36288)*a(n + 1) + (-13728*n^2 - 70032*n - 85824)*a(n + 2) + (6672*n^2 + 42312*n + 66936)*a(n + 3) + (-1325*n^2 - 10499*n - 20850)*a(n + 4) + (119*n^2 + 1148*n + 2772)*a(n + 5) + (-4*n^2 - 46*n - 132)*a(n + 6) = 0, a(0) = 1, a(1) = 3, a(2) = 16, a(3) = 95, a(4) = 590, a(5) = 3755},a(n), remember):

%p map(f, [$0..30]); # _Robert Israel_, Feb 12 2026

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

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

%Y Cf. A391459, A391465.

%Y Cf. A000129.

%K nonn

%O 0,2

%A _Seiichi Manyama_, Dec 10 2025