login

Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).

Expansion of e.g.f. 1/(exp(x) - x^4/24).
3

%I #18 Aug 21 2024 11:23:26

%S 1,-1,1,-1,2,-11,61,-281,1191,-5923,41791,-354091,2968021,-24059751,

%T 204718515,-1996937671,22125450621,-258434553861,3056858429581,

%U -37181421375349,482010195953821,-6741275765687821,99663246605243861,-1521712424934601901

%N Expansion of e.g.f. 1/(exp(x) - x^4/24).

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

%F a(n) ~ n! * 3*(-1)^n / ((1 + LambertW(3^(1/4) / 2^(5/4))) * 2^(2*n + 7) * LambertW(3^(1/4) / 2^(5/4))^(n+4)). - _Vaclav Kotesovec_, Mar 12 2022

%F a(n) = n! * Sum_{k=0..floor(n/4)} (-k-1)^(n-4*k)/(24^k*(n-4*k)!). - _Seiichi Manyama_, Aug 21 2024

%t m = 23; Range[0, m]! * CoefficientList[Series[1/(Exp[x] - x^4/24), {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^4/24)))

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

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

%Y Cf. A089148, A352309, A352310.

%Y Cf. A352304.

%K sign

%O 0,5

%A _Seiichi Manyama_, Mar 11 2022