A308331
a(n) = n! * [x^n] exp(exp(n*x)/(1 + x) - 1).
Original entry on oeis.org
1, 0, 3, 50, 1449, 61724, 3608515, 275520972, 26505128433, 3125830471928, 442286373458691, 73789189395157730, 14309059313820886681, 3186711239965235356776, 806772967716453793227523, 230153293624841114893344854, 73420355768107554901016231265
Offset: 0
-
Table[n! SeriesCoefficient[Exp[Exp[n x]/(1 + x) - 1], {x, 0, n}], {n, 0, 16}]
A347993
a(n) = n! * Sum_{k=1..n} (-1)^(k+1) * n^(n-k) / (n-k)!.
Original entry on oeis.org
1, 2, 15, 136, 1645, 24336, 426979, 8658560, 199234809, 5128019200, 145969492471, 4552809182208, 154404454932325, 5656950010320896, 222655633595044875, 9369696305273798656, 419790650812640438641, 19950175280765680680960, 1002394352017754098219999, 53092232229227200348160000
Offset: 1
-
Table[n! Sum[(-1)^(k + 1) n^(n - k)/(n - k)!, {k, 1, n}], {n, 1, 20}]
nmax = 20; CoefficientList[Series[-LambertW[-x]/(1 - LambertW[-x]^2), {x, 0, nmax}], x] Range[0, nmax]! // Rest
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a(n) = n! * sum(k=1, n, (-1)^(k+1)*n^(n-k)/(n-k)!); \\ Michel Marcus, Sep 23 2021
A347994
a(n) = n! * Sum_{k=1..n-1} (-1)^(k+1) * n^(n-k-2) / (n-k-1)!.
Original entry on oeis.org
0, 1, 4, 30, 296, 3720, 56652, 1014832, 20909520, 487198080, 12667470740, 363607605504, 11420819358456, 389646915374080, 14349217119054300, 567315485527234560, 23967624180805666208, 1077568488585047605248, 51369752823292604784420, 2588268388538639982592000
Offset: 1
-
Table[n! Sum[(-1)^(k + 1) n^(n - k - 2)/(n - k - 1)!, {k, 1, n - 1}], {n, 1, 20}]
nmax = 20; CoefficientList[Series[-LambertW[-x] - Log[1 - LambertW[-x]], {x, 0, nmax}], x] Range[0, nmax]! // Rest
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a(n) = n! * sum(k=1, n-1, (-1)^(k+1)*n^(n-k-2)/(n-k-1)!); \\ Michel Marcus, Sep 23 2021
A354944
a(n) = Sum_{k=0..n} binomial(n,k)^3 * k! * (-n)^(n-k).
Original entry on oeis.org
1, 0, -10, 60, 1560, -39880, -491760, 45672060, -155935360, -77656158000, 2116774828800, 166585352850620, -11925674437248000, -330617542587341880, 69148933431781898240, -543549949643024194500, -434534462104188331130880, 21521903478880966780355360
Offset: 0
-
Unprotect[Power]; 0^0 = 1; Table[Sum[Binomial[n, k]^3 k! (-n)^(n - k), {k, 0, n}], {n, 0, 17}]
Unprotect[Power]; 0^0 = 1; Table[n!^3 SeriesCoefficient[BesselI[0, 2 Sqrt[x]] Sum[(-n)^k x^k/k!^3, {k, 0, n}], {x, 0, n}], {n, 0, 17}]
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a(n) = sum(k=0, n, binomial(n,k)^3 * k! * (-n)^(n-k)); \\ Michel Marcus, Jun 12 2022
A336969
a(n) = n! * [x^n] 1 / (exp(n*x) - x).
Original entry on oeis.org
1, 0, -2, 33, -424, 495, 342864, -22382913, 915074432, -913039857, -5455432211200, 812138028148623, -75257247474017280, 1984517460320303415, 1155562494647499610112, -361521639388178369672625, 67461150715150454861692928, -6658374003334822571921759457
Offset: 0
-
Table[n! SeriesCoefficient[1/(Exp[n x] - x), {x, 0, n}], {n, 0, 17}]
Join[{1}, Table[n! Sum[(-n (n - k + 1))^k/k!, {k, 0, n}], {n, 1, 17}]]