A086331 -id:A086331 - cp's OEIS frontend

A362348 a(n) = n! * Sum_{k=0..floor(n/3)} k^k / (k! * (n-3*k)!).

1, 1, 1, 7, 25, 61, 1561, 10291, 40657, 1754425, 16632721, 90479071, 5469933481, 67591594357, 468224398825, 36386954606731, 554182030325281, 4663003095358321, 442756825853252257, 8014853488848923575, 79354642490200806841, 8901962495566386752941

Offset: 0

Seiichi Manyama, Apr 17 2023

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..427
Eric Weisstein's World of Mathematics, Lambert W-Function.

Cf. A086331, A362347, A362349.

my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(x)/(1+lambertw(-x^3))))

E.g.f.: exp(x) / (1 + LambertW(-x^3)).

a(n) ~ (exp(3*exp(-1/3)/2) + 2*cos(sqrt(3)*exp(-1/3)/2 - 2*Pi*n/3)) * n^n / (sqrt(3) * exp(2*n/3 + exp(-1/3)/2)). - Vaclav Kotesovec, Apr 18 2023

A277456 a(n) = 1 + Sum_{k=1..n} binomial(n,k) * 3^k * k^k.

Original entry on oeis.org

1, 4, 43, 847, 23881, 870721, 38894653, 2055873037, 125480383153, 8684069883409, 671922832985941, 57475677232902589, 5385592533714824521, 548596467532888667257, 60358911366712739334541, 7133453715771227363127301, 901261693601873814393568993

Offset: 0

Vaclav Kotesovec, Oct 16 2016

nonn

Robert Israel, Table of n, a(n) for n = 0..333

Cf. A086331, A277454.

[1] cat [1 + (&+[Binomial(n,k)*3^k*k^k: k in [1..n]]): n in [1..20]]; // G. C. Greubel, Sep 09 2018

f:= n -> 1 + add(binomial(n,k)*3^k*k^k,k=1..n):
map(f, [$0..20]); # Robert Israel, Oct 30 2016

Table[1 + Sum[Binomial[n, k]*3^k*k^k, {k, 1, n}], {n, 0, 20}]
CoefficientList[Series[E^x/(1+LambertW[-3*x]), {x, 0, 20}], x] * Range[0, 20]!

a(n) = 1 + sum(k=1, n, binomial(n,k) * 3^k * k^k); \\ Michel Marcus, Oct 30 2016

x='x+O('x^30); Vec(serlaplace(exp(x)/(1+lambertw(-3*x)))) \\ G. C. Greubel, Sep 09 2018

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

a(n) ~ exp(exp(-1)/3) * 3^n * n^n.

A277461 E.g.f.: sin(x)/(1+LambertW(-x)).

Original entry on oeis.org

0, 1, 2, 11, 104, 1241, 18216, 317715, 6414848, 147107953, 3776164000, 107253230171, 3339157316736, 113070818225353, 4137170839854976, 162653198951193059, 6837934005096620032, 306093463368534049761, 14535589272368159900160, 729835620496621069643179

Offset: 0

Vaclav Kotesovec, Oct 16 2016

nonn

G. C. Greubel, Table of n, a(n) for n = 0..385

Cf. A000312, A086331, A277462, A277463.

CoefficientList[Series[Sin[x]/(1+LambertW[-x]), {x, 0, 25}], x] * Range[0, 25]!
Table[Sin[Pi*n/2] + Sum[Binomial[n, k] * Sin[Pi*(n-k)/2] * k^k, {k, 1, n}], {n, 0, 25}] (* Vaclav Kotesovec, Oct 28 2016 *)

x = 'x + O('x^30); concat(0, Vec(serlaplace(sin(x)/(1+lambertw(-x))))) \\ Michel Marcus, Jun 12 2017

a(n) ~ sin(exp(-1)) * n^n.

A277464 Expansion of e.g.f. cosh(x)/(1 + LambertW(-x)).

Original entry on oeis.org

1, 1, 5, 30, 281, 3400, 50557, 890120, 18101617, 417464064, 10764826421, 306893014912, 9584448407305, 325407839778944, 11933432488693549, 470087171351873280, 19796492491889197025, 887518214183286390784, 42202928616264032249701, 2121583256369642798845952

Offset: 0

Vaclav Kotesovec, Oct 16 2016

nonn

G. C. Greubel, Table of n, a(n) for n = 0..385

Cf. A000312, A069856, A086331, A277462, A277463.

CoefficientList[Series[Cosh[x]/(1+LambertW[-x]), {x, 0, 25}], x] * Range[0, 25]!
Table[(1+(-1)^n + Sum[(1+(-1)^(n-k)) * Binomial[n,k] * k^k, {k, 1, n}])/2, {n, 0, 25}]

x='x+O('x^50); Vec(serlaplace(cosh(x)/(1 + lambertw(-x)))) \\ G. C. Greubel, Nov 07 2017

a(n) = sum(k=0, n\2, (n-2*k)^(n-2*k)*binomial(n, 2*k)); \\ Seiichi Manyama, Feb 15 2023

a(n) ~ cosh(exp(-1)) * n^n.

a(n) = Sum_{k=0..floor(n/2)} (n-2*k)^(n-2*k) * binomial(n,2*k). - Seiichi Manyama, Feb 15 2023

A362347 a(n) = n! * Sum_{k=0..floor(n/2)} k^k / (k! * (n-2*k)!).

Original entry on oeis.org

1, 1, 3, 7, 61, 261, 3991, 24403, 524217, 4149001, 114544171, 1111976031, 37492210933, 431097055117, 17165526306111, 228085258466731, 10472666396599921, 157882659583461393, 8211536252680154707, 138474928851961700791, 8045878340298511456941

Offset: 0

Seiichi Manyama, Apr 17 2023

nonn

Winston de Greef, Table of n, a(n) for n = 0..414
Eric Weisstein's World of Mathematics, Lambert W-Function.

Cf. A086331, A362348, A362349.

Cf. A277614.

my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(x)/(1+lambertw(-x^2))))

E.g.f.: exp(x) / (1 + LambertW(-x^2)).

a(n) ~ (exp(2*exp(-1/2)) + (-1)^n) * n^n / (sqrt(2) * exp(n/2 + exp(-1/2))). - Vaclav Kotesovec, Aug 05 2025

A277454 a(n) = 1 + Sum_{k=1..n} binomial(n,k) * 2^k * k^k.

Original entry on oeis.org

1, 3, 21, 271, 5065, 122811, 3651997, 128566663, 5227782161, 241072839667, 12430169195941, 708612945554559, 44253858433505497, 3004570398043291819, 220341964157226260525, 17357760973540312138231, 1461813975265547356467745, 131061164660246579394042339

Offset: 0

Vaclav Kotesovec, Oct 16 2016

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..351

Cf. A086331, A277456.

Table[1+Sum[Binomial[n, k]*2^k*k^k, {k, 1, n}], {n, 0, 20}]
CoefficientList[Series[E^x/(1+LambertW[-2*x]), {x, 0, 20}], x] * Range[0, 20]!

{a(n) = sum(k=0, n, binomial(n, k)*(2*k)^k)} \\ Seiichi Manyama, Jan 12 2019

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

a(n) ~ exp(exp(-1)/2) * 2^n * n^n.

A277463 E.g.f.: sinh(x)/(1+LambertW(-x)).

Original entry on oeis.org

0, 1, 2, 13, 112, 1321, 19296, 335637, 6764864, 154946449, 3973820800, 112789880413, 3509627281920, 118790978349369, 4344883388878592, 170767066282574821, 7177162988688031744, 321206181612447781921, 15250250261039350358016, 765586309042945067185581

Offset: 0

Vaclav Kotesovec, Oct 16 2016

nonn

G. C. Greubel, Table of n, a(n) for n = 0..385

Cf. A000312, A069856, A086331, A277461, A277464.

CoefficientList[Series[Sinh[x]/(1+LambertW[-x]), {x, 0, 25}], x] * Range[0, 25]!
Table[(1-(-1)^n + Sum[(1-(-1)^(n-k)) * Binomial[n,k] * k^k, {k, 1, n}])/2, {n, 0, 25}]

x='x+O('x^50); concat([0], Vec(serlaplace(sinh(x)/(1 + lambertw(-x))))) \\ G. C. Greubel, Nov 05 2017

a(n) ~ sinh(exp(-1)) * n^n.

A277468 E.g.f.: tanh(x)/(1+LambertW(-x)).

Original entry on oeis.org

0, 1, 2, 10, 100, 1216, 17766, 309744, 6260360, 143641600, 3688352650, 104786813440, 3263080663404, 110514370068480, 4044232154193518, 159019302501971968, 6685886706336107536, 299315231931854749696, 14214873507079452102162, 713784039156929684963328

Offset: 0

Vaclav Kotesovec, Oct 16 2016

nonn

G. C. Greubel, Table of n, a(n) for n = 0..385

Cf. A000312, A086331, A277467.

CoefficientList[Series[Tanh[x]/(1+LambertW[-x]), {x, 0, 25}], x] * Range[0, 25]!
Flatten[{0, Table[2^(n+1)*(2^(n+1) - 1)*BernoulliB[n+1]/(n+1) + Sum[Binomial[n, k]*2^(k+1)*(2^(k+1) - 1) * BernoulliB[k+1]/(k+1)*(n-k)^(n-k), {k, 1, n-1}], {n, 1, 25}]}] (* Vaclav Kotesovec, Oct 28 2016 *)

x='x+O('x^50); concat([0], Vec(serlaplace(tanh(x)/(1 + lambertw(-x))))) \\ G. C. Greubel, Nov 05 2017

a(n) ~ tanh(exp(-1)) * n^n.

A336214 a(n) = Sum_{k=0..n} k^n * binomial(n,k)^n, with a(0)=1.

Original entry on oeis.org

1, 1, 8, 270, 41984, 30706250, 94770093312, 1336016204844832, 76829717664330940416, 19838680914222199482800274, 20521247958509575370600000000000, 94285013320530947020636486516362047300, 1715947732437668013396578734960052732361179136

Offset: 0

Vaclav Kotesovec, Jul 12 2020

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..56

Cf. A072034, A086331, A167008, A167010, A328812, A336188, A336204, A336212, A336213.

Flatten[{1, Table[Sum[k^n*Binomial[n, k]^n, {k, 1, n}], {n, 1, 15}]}]

a(n) = if (n==0, 1, sum(k=0, n, k^n * binomial(n,k)^n)); \\ Michel Marcus, Jul 13 2020

a(n) ~ c * exp(-1/4) * 2^(n^2 - n/2) * n^(n/2) / Pi^(n/2), where c = Sum_{k = -infinity..infinity} exp(-2*k*(k-1)) = exp(1/2) * sqrt(Pi/2) * EllipticTheta(3, -Pi/2, exp(-Pi^2/2)) = 2.036643566277677716389243890291939003151565... if n is even and c = Sum_{k = -infinity..infinity} exp(-2*k^2 + 1/2) = exp(1/2) * EllipticTheta(3, 0, exp(-2)) = 2.096087809957308346119920713317351288828811... if n is odd.

a(n) = n^n * A328812(n-1) for n > 0. - Seiichi Manyama, Jul 15 2020

A355494 Expansion of Sum_{k>=0} (k * x/(1 - x))^k.

Original entry on oeis.org

1, 1, 5, 36, 350, 4328, 65132, 1155904, 23640724, 547544032, 14166236708, 404944248104, 12674392793900, 431104742439088, 15834117059443828, 624575921756875960, 26332801242942780668, 1181750740315156943936, 56244454481507648435012

Offset: 0

Seiichi Manyama, Jul 04 2022

nonn

Winston de Greef, Table of n, a(n) for n = 0..385

Cf. A355495, A355496.

Cf. A086331.

Join[{1}, Table[Sum[k^k * Binomial[n-1,k-1], {k,1,n}], {n, 1, 20}]] (* Vaclav Kotesovec, Jul 05 2022 *)

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

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

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

a(n) ~ exp(exp(-1)) * n^n. - Vaclav Kotesovec, Jul 05 2022

cp's OEIS Frontend

A362348 a(n) = n! * Sum_{k=0..floor(n/3)} k^k / (k! * (n-3*k)!).

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A277456 a(n) = 1 + Sum_{k=1..n} binomial(n,k) * 3^k * k^k.

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A277461 E.g.f.: sin(x)/(1+LambertW(-x)).

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A277464 Expansion of e.g.f. cosh(x)/(1 + LambertW(-x)).

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A362347 a(n) = n! * Sum_{k=0..floor(n/2)} k^k / (k! * (n-2*k)!).

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A277454 a(n) = 1 + Sum_{k=1..n} binomial(n,k) * 2^k * k^k.

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Mathematica

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A277463 E.g.f.: sinh(x)/(1+LambertW(-x)).

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Mathematica

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A277468 E.g.f.: tanh(x)/(1+LambertW(-x)).

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