A358741 -id:A358741 - cp's OEIS frontend

A358738 Expansion of Sum_{k>=0} k! * ( x/(1 - k*x) )^k.

1, 1, 3, 15, 103, 893, 9341, 114355, 1603155, 25318137, 444689497, 8597568671, 181430298479, 4149361409077, 102229328244837, 2699254206069387, 76038064580742091, 2276259442660623857, 72160287650141753777, 2414950992007231422007

Offset: 0

Seiichi Manyama, Nov 29 2022

nonn

Cf. A001339, A006153, A080108.

Cf. A358740, A358741.

nmax = 20; CoefficientList[Series[Sum[k! * (x/(1 - k*x))^k, {k, 0, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Feb 18 2023 *)

my(N=20, x='x+O('x^N)); Vec(sum(k=0, N, k!*(x/(1-k*x))^k))

a(n) = if(n==0, 1, sum(k=1, n, k!*k^(n-k)*binomial(n-1, k-1)));

a(n) = Sum_{k=1..n} k! * k^(n-k) * binomial(n-1,k-1) for n > 0.

a(n) ~ n! / ((1 + LambertW(1))^2 * LambertW(1)^n). - Vaclav Kotesovec, Feb 18 2023

A358740 Expansion of Sum_{k>=0} k! * ( k * x/(1 - k*x) )^k.

Original entry on oeis.org

1, 1, 9, 195, 7699, 482309, 43994741, 5508667927, 906931827831, 189998213001033, 49359340639141993, 15573690455085072011, 5866304418414451865723, 2600416934781350100016717, 1340037064604153376788884701, 794358527033920600533985973631

Offset: 0

Seiichi Manyama, Nov 29 2022

nonn

Cf. A358738, A358741.

Cf. A195242.

nmax = 20; CoefficientList[1 + Series[Sum[k! * (k * x/(1 - k*x))^k, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Feb 18 2023 *)

my(N=20, x='x+O('x^N)); Vec(sum(k=0, N, k!*(k*x/(1-k*x))^k))

a(n) = if(n==0, 1, sum(k=1, n, k!*k^n*binomial(n-1, k-1)));

a(n) = Sum_{k=1..n} k! * k^n * binomial(n-1,k-1) for n > 0.

a(n) ~ exp(exp(-1)) * n! * n^n. - Vaclav Kotesovec, Feb 18 2023

cp's OEIS Frontend

A358738 Expansion of Sum_{k>=0} k! * ( x/(1 - k*x) )^k.

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A358740 Expansion of Sum_{k>=0} k! * ( k * x/(1 - k*x) )^k.

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Author

Keywords

Crossrefs

Programs

Mathematica

PARI

PARI

Formula