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Expansion of e.g.f. 1/(2 - x^2 - exp(x)).
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%I #28 Apr 17 2022 20:53:06

%S 1,1,5,25,195,1781,20043,260317,3881083,64978861,1209674883,

%T 24764370533,553130762451,13383468009445,348741065652619,

%U 9736370899180813,289948812396124875,9174320178178480829,307362076657095903411,10869452423023391315413,404614540610985119535715

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

%H Seiichi Manyama, <a href="/A346269/b346269.txt">Table of n, a(n) for n = 0..408</a>

%F E.g.f.: 1/(2 - x^2 - exp(x)).

%F a(n) ~ n! / ((2 + 2*r - r^2) * r^(n+1)), where r = A201752 = 0.5372744491738566... is the positive root of the equation 2 - r^2 - exp(r) = 0.

%F a(0) = a(1) = 1; a(n) = n * (n-1) * a(n-2) + Sum_{k=1..n} binomial(n,k) * a(n-k). - _Seiichi Manyama_, Mar 11 2022

%t nmax = 20; CoefficientList[Normal[Series[1/(2-x^2-E^x), {x, 0, nmax}]], x] * Range[0, nmax]!

%o (PARI) my(x='x+O('x^25)); Vec(serlaplace(1/(2 - x^2 - exp(x)))) \\ _Michel Marcus_, Jul 12 2021

%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, 2); \\ _Seiichi Manyama_, Mar 12 2022

%Y Cf. A006155, A201752.

%K nonn

%O 0,3

%A _Vaclav Kotesovec_, Jul 12 2021