A352304 - OEIS

%I #19 Aug 21 2024 11:23:39

%S 1,-1,1,-1,25,-241,1441,-6721,67201,-1185409,16652161,-180639361,

%T 2098673281,-37526586241,785718950017,-14516030954881,247504017895681,

%U -4832929862019841,116556246644716801,-2930255897793414913,69746855593499124481,-1673960044278244020481

%N Expansion of e.g.f. 1/(exp(x) - x^4).

%F a(n) = n * (n-1) * (n-2) * (n-3) * a(n-4) - Sum_{k=1..n} binomial(n,k) * a(n-k) for n > 3.

%F a(n) ~ n! * (-1)^n / ((1 + LambertW(1/4)) * 2^(2*n + 10) * LambertW(1/4)^(n+4)). - _Vaclav Kotesovec_, Mar 12 2022

%F a(n) = n! * Sum_{k=0..floor(n/4)} (-k-1)^(n-4*k)/(n-4*k)!. - _Seiichi Manyama_, Aug 21 2024

%t m = 21; Range[0, m]! * CoefficientList[Series[1/(Exp[x] - x^4), {x, 0, m}], x] (* _Amiram Eldar_, Mar 12 2022 *)

%o (PARI) my(N=40, x='x+O('x^N)); Vec(serlaplace(1/(exp(x)-x^4)))

%o (PARI) b(n, m) = if(n==0, 1, sum(k=1, n, (-1+(k==m)*m!)*binomial(n, k)*b(n-k, m)));

%o a(n) = b(n, 4);

%Y Cf. A089148, A352302, A352303.

%Y Cf. A352300, A352311.

%K sign

%O 0,5

%A _Seiichi Manyama_, Mar 11 2022