A391461 - OEIS

%I #15 Feb 13 2026 10:25:25

%S 1,2,11,66,412,2630,17029,111344,733202,4854062,32269865,215245074,

%T 1439609582,9650033224,64808040028,435935055728,2936358225450,

%U 19802076360062,133678410354017,903246297123430,6108010982206984,41333625143684982,279888021880711789

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

%H Robert Israel, <a href="/A391461/b391461.txt">Table of n, a(n) for n = 0..1193</a>

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

%F D-finite with recurrence: (-4752*n^2 - 9504*n - 4224)*a(n) + (6104*n^2 + 15208*n + 9408)*a(n + 1) + (-2042*n^2 - 7516*n - 6672)*a(n + 2) + (265*n^2 + 1337*n + 1608)*a(n + 3) + (-12*n^2 - 78*n - 120)*a(n + 4) = 0. - _Robert Israel_, Feb 12 2026

%p f:= gfun:-rectoproc({(-4752*n^2 - 9504*n - 4224)*a(n) + (6104*n^2 + 15208*n + 9408)*a(n + 1) + (-2042*n^2 - 7516*n - 6672)*a(n + 2) + (265*n^2 + 1337*n + 1608)*a(n + 3) + (-12*n^2 - 78*n - 120)*a(n + 4), a(0) = 1, a(1) = 2, a(2) = 11, a(3) = 66, a(4) = 412}, 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+1)*binomial(3*n, n-k))/n);

%Y Cf. A289684, A391464.

%Y Cf. A000129, A006013.

%K nonn

%O 0,2

%A _Seiichi Manyama_, Dec 10 2025