A346411
a(n) = (n!)^2 * Sum_{k=0..n-1} (-1)^k / ((n-k) * k!)^2.
Original entry on oeis.org
0, 1, -3, 4, -8, 1, 353, 27224, 1871840, 147012849, 13684928021, 1514370713340, 197964773810648, 30300949591876913, 5380510834911767033, 1098630080602791984784, 255851291397441057781120, 67450889282916741495608737, 19994198644782014829579657837, 6623096362909598587714211804212
Offset: 0
-
Table[(n!)^2 Sum[(-1)^k/((n - k) k!)^2, {k, 0, n - 1}], {n, 0, 19}]
nmax = 19; CoefficientList[Series[PolyLog[2, x] BesselJ[0, 2 Sqrt[x]], {x, 0, nmax}], x] Range[0, nmax]!^2
A346427
E.g.f.: -log(1 - log(1 + x) * exp(x)).
Original entry on oeis.org
0, 1, 2, 7, 29, 183, 1319, 12122, 124802, 1508581, 20150509, 302637564, 4960500764, 89164162579, 1730245993111, 36241995276276, 812108432244304, 19430625834864633, 493622198791114665, 13283773364613034324, 377224137563670860492, 11278211794764786428831
Offset: 0
-
nmax = 21; CoefficientList[Series[-Log[1 - Log[1 + x] Exp[x]], {x, 0, nmax}], x] Range[0, nmax]!
A002741[n_] := A002741[n] = n! Sum[(-1)^k/((n - k) k!), {k, 0, n - 1}]; a[0] = 0; a[n_] := a[n] = -(-1)^n A002741[n] - (1/n) Sum[(-1)^(n - k) Binomial[n, k] A002741[n - k] k a[k], {k, 1, n - 1}]; Table[a[n], {n, 0, 21}]
-
my(x='x+O('x^25)); concat(0, Vec(serlaplace(-log(1 - log(1+x) * exp(x))))) \\ Michel Marcus, Jul 19 2021
A351502
Expansion of e.g.f. 1/(1 + log(1 - x)*exp(-x)).
Original entry on oeis.org
1, 1, 1, 2, 10, 59, 373, 2736, 23504, 229029, 2477219, 29473344, 383104588, 5401356583, 82069677701, 1336740758544, 23234632127072, 429259519490985, 8399672396793063, 173538299521211128, 3774815414843398588, 86230662745426403771, 2063931187442813081881
Offset: 0
-
With[{nn=30},CoefficientList[Series[1/(1+Log[1-x]Exp[-x]),{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, May 03 2023 *)
-
my(N=30, x='x+O('x^N)); Vec(serlaplace(1/(1+log(1-x)*exp(-x))))
A352150
a(n) = Sum_{k=0..n-1} (-1)^k * binomial(n,k)^2 * (n-k-1)!.
Original entry on oeis.org
0, 1, -3, 2, -6, -1, -5, 132, 1624, 17145, 174509, 1789842, 18659146, 196678143, 2057524963, 20460314396, 171030108768, 529697015489, -27050118799923, -1079945984126798, -30289996673371254, -765129844741436785, -18575997643525737477, -444653043972658034044
Offset: 0
-
Table[Sum[(-1)^k Binomial[n, k]^2 (n - k - 1)!, {k, 0, n - 1}], {n, 0, 23}]
nmax = 23; Assuming[x > 0, CoefficientList[Series[BesselJ[0, 2 Sqrt[x]] (ExpIntegralEi[x] - Log[x] - EulerGamma), {x, 0, nmax}], x]] Range[0, nmax]!^2
-
a(n) = sum(k=0, n-1, (-1)^k * binomial(n,k)^2 * (n-k-1)!); \\ Michel Marcus, Mar 06 2022
A306948
Expansion of e.g.f. (1 + x)*log(1 + x)*exp(x).
Original entry on oeis.org
0, 1, 3, 5, 8, 9, 19, -15, 216, -1407, 11803, -108483, 1106192, -12363703, 150381243, -1977666743, 27965386320, -423158076351, 6822782712723, -116781368777867, 2114916140765496, -40404117909336247, 812091479233464131, -17130720178674680031, 378423227774537955688
Offset: 0
-
a:=series((1 + x)*log(1 + x)*exp(x),x=0,25): seq(n!*coeff(a, x, n), n=0..24); # Paolo P. Lava, Mar 26 2019
-
nmax = 24; CoefficientList[Series[(1 + x) Log[1 + x] Exp[x], {x, 0, nmax}], x] Range[0, nmax]!
Table[Sum[StirlingS1[n, k] BellB[k] k, {k, 0, n}], {n, 0, 24}]
Table[Sum[(-1)^(k - 1) Binomial[n, k] (n - k + 1) (k - 1)!, {k, 1, n}], {n, 0, 24}]
A334258
a(n) = (-1)^n * exp(n) * Sum_{k>=1} (-1)^k * n^(k-1) * k^n / k!.
Original entry on oeis.org
1, 1, 1, -5, -74, -679, -4899, -17289, 325837, 10627109, 199348590, 2684041427, 15872610469, -546948563407, -27499774835519, -778467357484561, -15311413773551790, -125363405319188419, 6452292137017871097, 436442148982835915339, 16494863323310244977581
Offset: 1
-
nmax = 21; CoefficientList[InverseSeries[Series[-Log[1 - x] Exp[-x], {x, 0, nmax}], x], x] Range[0, nmax]! // Rest
Table[Sum[(-1)^(n - k) StirlingS2[n, k] n^(k - 1), {k, 1, n}], {n, 1, 21}]
Table[(-1)^n BellB[n, -n]/n, {n, 1, 21}]
-
a(n) = sum(k=1, n, (-1)^(n-k) * stirling(n,k,2) * n^(k-1)); \\ Michel Marcus, Apr 20 2020