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a(n) = exp(-n) * Sum_{k>=0} n^k * (k - 1)^n / k!.
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3
1, 0, 3, 29, 397, 6879, 144751, 3587100, 102351929, 3305310065, 119186370091, 4746969337923, 206966647324933, 9804683604806908, 501491905963040903, 27544070654283355889, 1616869985889305862385, 101020181695996141703335, 6693303018177050431484035, 468770856837303230888704208
FORMULA
a(n) = n! * [x^n] exp(n*(exp(x) - 1) - x).
a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(n,k) * BellPolynomial_k(n).
MATHEMATICA
Table[n! SeriesCoefficient[Exp[n (Exp[x] - 1) - x], {x, 0, n}], {n, 0, 19}]
Table[Sum[(-1)^(n - k) Binomial[n, k] BellB[k, n], {k, 0, n}], {n, 0, 19}]
Expansion of e.g.f. exp(2 * (1 - exp(-x)) + x).
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3
1, 3, 7, 11, 7, -5, 23, 75, -281, -101, 4663, -14229, -41721, 532667, -1464489, -8840053, 103689511, -313202725, -2348557705, 32041266859, -127039882425, -762423051013, 14393151011735, -81523161874741, -236027974047897, 8564406463119387
FORMULA
a(n) = exp(2) * (-1)^n * Sum_{k>=0} (-2)^k * (k - 1)^n / k!.
a(0) = 1; a(n) = a(n-1) + 2 * Sum_{k=0..n-1} (-1)^(n-k-1) * binomial(n-1,k) * a(k).
MATHEMATICA
nmax = 25; CoefficientList[Series[Exp[2 (1 - Exp[-x]) + x], {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = a[n - 1] + 2 Sum[(-1)^(n - k - 1) Binomial[n - 1, k] a[k], {k, 0, n - 1}]; Table[a[n], {n, 0, 25}]
PROG
(PARI) my(N=33, x='x+O('x^N)); Vec(serlaplace(exp(2 * (1 - exp(-x)) + x))) \\ Joerg Arndt, Jul 04 2020
Expansion of e.g.f. exp(3 * (1 - exp(-x)) + x).
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3
1, 4, 13, 31, 40, -23, -95, 490, 823, -8393, 3766, 174775, -658787, -2751404, 34033297, -55552037, -1170734432, 9362348365, 3277050925, -562286419646, 3848880970147, 8815342530739, -356804325202730, 2389771436686339, 8677476137729929, -302470260552857660
FORMULA
a(n) = exp(3) * (-1)^n * Sum_{k>=0} (-3)^k * (k - 1)^n / k!.
a(0) = 1; a(n) = a(n-1) + 3 * Sum_{k=0..n-1} (-1)^(n-k-1) * binomial(n-1,k) * a(k).
MATHEMATICA
nmax = 25; CoefficientList[Series[Exp[3 (1 - Exp[-x]) + x], {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = a[n - 1] + 3 Sum[(-1)^(n - k - 1) Binomial[n - 1, k] a[k], {k, 0, n - 1}]; Table[a[n], {n, 0, 25}]
Expansion of e.g.f. exp(4 * (1 - exp(-x)) + x).
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3
1, 5, 21, 69, 149, 69, -619, -187, 9365, -3515, -193643, 453957, 4704917, -29425595, -83918443, 1640246085, -3184430955, -74516517307, 604223657877, 1324972362053, -52526078298475, 264984579390533, 2477371363954069, -44206576595187899, 133280843118435477
FORMULA
a(n) = exp(4) * (-1)^n * Sum_{k>=0} (-4)^k * (k - 1)^n / k!.
a(0) = 1; a(n) = a(n-1) + 4 * Sum_{k=0..n-1} (-1)^(n-k-1) * binomial(n-1,k) * a(k).
MATHEMATICA
nmax = 24; CoefficientList[Series[Exp[4 (1 - Exp[-x]) + x], {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = a[n - 1] + 4 Sum[(-1)^(n - k - 1) Binomial[n - 1, k] a[k], {k, 0, n - 1}]; Table[a[n], {n, 0, 24}]
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