A346269 Expansion of e.g.f. 1/(2 - x^2 - exp(x)).

1, 1, 5, 25, 195, 1781, 20043, 260317, 3881083, 64978861, 1209674883, 24764370533, 553130762451, 13383468009445, 348741065652619, 9736370899180813, 289948812396124875, 9174320178178480829, 307362076657095903411, 10869452423023391315413, 404614540610985119535715

Offset: 0

Author

Vaclav Kotesovec, Jul 12 2021

Keywords

• nonn

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..408

Crossrefs

Cf. A006155, A201752.

Programs

Mathematica

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

PARI

my(x='x+O('x^25)); Vec(serlaplace(1/(2 - x^2 - exp(x)))) \\ Michel Marcus, Jul 12 2021

PARI

b(n, m) = if(n==0, 1, sum(k=1, n, (1+(k==m)*m!)*binomial(n, k)*b(n-k, m)));
a(n) = b(n, 2); \\ Seiichi Manyama, Mar 12 2022

Formula

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

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.

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