A334670 - OEIS

%I #33 Apr 29 2021 04:33:44

%S 0,1,8,71,744,9129,129072,2071215,37237680,741975345,16236211320,

%T 387182170935,9995788416600,277792140828825,8269430130712800,

%U 262542617405726175,8855805158351474400,316285840413064454625,11924219190760084593000,473245342972281190686375,19722890048636406588957000

%N a(n) = (2*n+1)!! * (Sum_{k=1..n} 1/(2*k+1)).

%H Seiichi Manyama, <a href="/A334670/b334670.txt">Table of n, a(n) for n = 0..403</a>

%F a(n) + A001147(n+1) = A004041(n).

%F a(n) = (2*n+1) * a(n-1) + A001147(n) for n>0.

%F P-finite with recurrence a(n) = 4*n*a(n-1) - (2*n-1)^2 * a(n-2) for n>1.

%e a(1) = 3 * (1/3) = 1.

%e a(2) = 3*5 * (1/3 + 1/5) = 8.

%e a(3) = 3*5*7 * (1/3 +1/5 + 1/7) = 71.

%t a[n_] := (2*n + 1)!! * Sum[1/(2*k + 1), {k, 1, n}]; Array[a, 21, 0] (* _Amiram Eldar_, Apr 29 2021 *)

%o (PARI) {a(n) = prod(k=1, n, 2*k+1)*sum(k=1, n, 1/(2*k+1))}

%o (PARI) {a(n) = if(n<2, n, 4*n*a(n-1)-(2*n-1)^2*a(n-2))}

%Y Column k=1 of A335095.

%Y Cf. A001147, A001705, A004041, A288875, A334000, A334066.

%K nonn

%O 0,3

%A _Seiichi Manyama_, Sep 10 2020