A352309 - OEIS

%I #19 Aug 21 2024 11:22:52

%S 1,-1,2,-7,31,-171,1141,-8863,78653,-785557,8716861,-106395741,

%T 1416724915,-20436548575,317477947151,-5284248213091,93816998697721,

%U -1769737117839849,35347571931577609,-745232024035027225,16538641134235561631,-385387334950748244451

%N Expansion of e.g.f. 1/(exp(x) - x^2/2).

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

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

%F a(n) = n! * Sum_{k=0..floor(n/2)} (-k-1)^(n-2*k)/(2^k*(n-2*k)!). - _Seiichi Manyama_, Aug 21 2024

%t m = 21; Range[0, m]! * CoefficientList[Series[1/(Exp[x] - x^2/2), {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^2/2)))

%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, 2);

%Y Cf. A089148, A352310, A352311.

%Y Cf. A352302, A352306.

%K sign

%O 0,3

%A _Seiichi Manyama_, Mar 11 2022