A395202 - OEIS

%I #11 Apr 16 2026 09:59:36

%S 1,21,849,61941,7214529,1229315061,288483011409,89219766504981,

%T 35169525934911489,17212551024649711701,10240591076060907554769,

%U 7278858609414500847895221,6091844983487415298481901249,5929573147218620069431200095541,6641729004270943179521683072030929

%N a(n) = (2*n)! * [x^(2*n)] (f(x)^3 + f(-x)^3)/2 and f(x) = 1/(cosh(x) - sqrt(2) * sinh(x)).

%F a(n) = (2*n)! * [x^(2*n)] cosh(x) * (1 + 7*sinh(x)^2)/(1 - sinh(x)^2)^3.

%F Let A(n,k) = (2*n)! * [x^(2*n)] (f(x)^k + f(-x)^k)/2. A(0,k) = 1 and A(n,k) = k*(k+1) * A(n-1,k+2) + k^2 * A(n-1,k) for n > 0. a(n) = A(n,3).

%o (PARI) a(n) = my(x='x+O('x^(2*n+1))); (2*n)!*polcoef(cosh(x)*(1+7*sinh(x)^2)/(1-sinh(x)^2)^3, 2*n);

%Y Column k=3 of A395200.

%K nonn

%O 0,2

%A _Seiichi Manyama_, Apr 15 2026