A277508
E.g.f.: -1/((1-LambertW(-x))*(1+x)).
Original entry on oeis.org
-1, 2, -4, 15, -44, 385, -294, 32473, 280120, 8528049, 170757910, 4748977321, 132530188308, 4210910824393, 142940443542274, 5273270156096265, 208276214340505456, 8800344095155520353, 395536292522024420142, 18853858817276143333321, 949787282108877829653580
Offset: 0
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CoefficientList[Series[-1/(1-LambertW[-x])/(1+x), {x, 0, 20}], x] * Range[0, 20]!
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x='x+O('x^50); Vec(serlaplace(-1/((1 - lambertw(-x))*(1+x)))) \\ G. C. Greubel, Nov 08 2017
A354437
a(n) = n! * Sum_{k=0..n} (-k)^(n-k)/k!.
Original entry on oeis.org
1, 1, -1, 1, 13, -199, 2251, -19991, 7001, 7530193, -330734249, 11005284401, -300961551131, 4886902605001, 184195977487523, -28517140157423399, 2322376314679777201, -153646291657993064671, 8388000381774954552751, -287686436757241322569247
Offset: 0
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Join[{1}, Table[n!*Sum[ (-k)^(n - k)/k!, {k, 0, n}], {n, 1, 20}]] (* Vaclav Kotesovec, May 28 2022 *)
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a(n) = n!*sum(k=0, n, (-k)^(n-k)/k!);
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my(N=20, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, x^k/(k!*(1+k*x)))))
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from math import factorial
def A354437(n): return sum(factorial(n)*(-k)**(n-k)//factorial(k) for k in range(n+1)) # Chai Wah Wu, May 28 2022
A360596
Expansion of e.g.f. 1/( (1 - x) * (1 + LambertW(-2*x)) ).
Original entry on oeis.org
1, 3, 22, 282, 5224, 126120, 3742704, 131612432, 5347866752, 246490091136, 12704900911360, 724072211436288, 45209213973292032, 3068872654856532992, 225023336997933996032, 17724257054969009940480, 1492513932494133333753856, 133800772458366199028023296
Offset: 0
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my(N=20, x='x+O('x^N)); Vec(serlaplace(1/((1-x)*(1+lambertw(-2*x)))))
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a(n) = n!*sum(k=0, n, (2*k)^k/k!);
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a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=i*v[i]+(2*i)^i); v;
A362682
Expansion of e.g.f. exp(-LambertW(-x))/(1+x).
Original entry on oeis.org
1, 0, 3, 7, 97, 811, 11941, 178557, 3354513, 69809383, 1659853861, 43658971753, 1268252733001, 40206626846283, 1383302292511413, 51308059650256741, 2041494097105707937, 86734131445797216847, 3919172491760452282693, 187679722656551406628833
Offset: 0
Showing 1-4 of 4 results.