A336949 -id:A336949 - cp's OEIS frontend

A336948 E.g.f.: 1 / (exp(-3*x) - x).

1, 4, 23, 195, 2229, 31863, 546255, 10925757, 249753897, 6422808411, 183524701779, 5768419379913, 197791542799965, 7347180526444359, 293912722687075767, 12597352573293062757, 575928946256877156177, 27976119070974574461363, 1438896686251112024068251

Offset: 0

Ilya Gutkovskiy, Aug 08 2020

nonn

Cf. A072597, A328182, A336947, A336949, A336951.

nmax = 18; CoefficientList[Series[1/(Exp[-3 x] - x), {x, 0, nmax}], x] Range[0, nmax]!
Table[n! Sum[(3 (n - k + 1))^k/k!, {k, 0, n}], {n, 0, 18}]
a[0] = 1; a[n_] := a[n] = 4 n a[n - 1] - Sum[Binomial[n, k] (-3)^k a[n - k], {k, 2, n}]; Table[a[n], {n, 0, 18}]

seq(n)={ Vec(serlaplace(1 / (exp(-3*x + O(x*x^n)) - x))) } \\ Andrew Howroyd, Aug 08 2020

a(n) = n! * Sum_{k=0..n} (3 * (n-k+1))^k / k!.
a(0) = 1; a(n) = 4 * n * a(n-1) - Sum_{k=2..n} binomial(n,k) * (-3)^k * a(n-k).
a(n) ~ n! / ((1 + LambertW(3)) * (LambertW(3)/3)^(n+1)). - Vaclav Kotesovec, Aug 09 2021

A336947 E.g.f.: 1 / (exp(-2*x) - x).

Original entry on oeis.org

1, 3, 14, 98, 920, 10792, 151888, 2494032, 46803072, 988095104, 23178247424, 598074306304, 16835199087616, 513385352524800, 16859837094942720, 593234633904293888, 22265289445252628480, 887889931920920313856, 37489832605652634763264, 1670894259596134872711168

Offset: 0

Ilya Gutkovskiy, Aug 08 2020

nonn

Cf. A072597, A216794, A336948, A336949, A336950.

nmax = 19; CoefficientList[Series[1/(Exp[-2 x] - x), {x, 0, nmax}], x] Range[0, nmax]!
Table[n! Sum[(2 (n - k + 1))^k/k!, {k, 0, n}], {n, 0, 19}]
a[0] = 1; a[n_] := a[n] = 3 n a[n - 1] - Sum[Binomial[n, k] (-2)^k a[n - k], {k, 2, n}]; Table[a[n], {n, 0, 19}]

seq(n)={ Vec(serlaplace(1 / (exp(-2*x + O(x*x^n)) - x))) } \\ Andrew Howroyd, Aug 08 2020

a(n) = n! * Sum_{k=0..n} (2 * (n-k+1))^k / k!.
a(0) = 1; a(n) = 3 * n * a(n-1) - Sum_{k=2..n} binomial(n,k) * (-2)^k * a(n-k).
a(n) ~ n! / ((1 + LambertW(2)) * (LambertW(2)/2)^(n+1)). - Vaclav Kotesovec, Aug 09 2021

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

Ilya Gutkovskiy, Aug 09 2020

sign

Cf. A089148, A134095, A302398, A336949, A336958.

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}]]

a(n) = n! * Sum_{k=0..n} (-n * (n-k+1))^k / k!.

cp's OEIS Frontend

A336948 E.g.f.: 1 / (exp(-3*x) - x).

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A336947 E.g.f.: 1 / (exp(-2*x) - x).

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Mathematica

PARI

Formula

A336969 a(n) = n! * [x^n] 1 / (exp(n*x) - x).

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Mathematica

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