A277473 -id:A277473 - cp's OEIS frontend

A277483 E.g.f.: -arcsin(x)*LambertW(-x).

0, 0, 2, 6, 40, 340, 3984, 57050, 982528, 19616328, 446355840, 11384327438, 321701896704, 9973046260060, 336499112011776, 12274383608508450, 481282311712489472, 20185816487436968208, 901732370496365076480, 42742176871086712813974, 2142556308913381810012160

Offset: 0

Vaclav Kotesovec, Oct 17 2016

nonn

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

Cf. A000169, A277469, A277473, A277484.

CoefficientList[Series[-ArcSin[x]*LambertW[-x], {x, 0, 20}], x] * Range[0, 20]!
Flatten[{0, Table[Sum[Binomial[n, k] * (1-(-1)^k)/2 * (k-2)!!^2 * (n-k)^(n-k-1), {k, 1, n-1}], {n, 1, 20}]}] (* Vaclav Kotesovec, Oct 28 2016 *)

x='x+O('x^50); concat([0,0], Vec(serlaplace(- asin(x)*lambertw(-x) ))) \\ G. C. Greubel, Nov 08 2017

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

A277485 E.g.f.: -exp(2x)LambertW(-x).

Original entry on oeis.org

0, 1, 6, 33, 216, 1865, 21228, 303765, 5222864, 104540337, 2383558740, 60933722069, 1725392415288, 53590463856345, 1811281159509500, 66172416761172885, 2598298697830360992, 109116931783034360801, 4880122696811960470692, 231565260558289051906965

Offset: 0

Vaclav Kotesovec, Oct 17 2016

nonn

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

Cf. A000169, A069856, A086331, A277457, A277473, A277474.

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

x='x+O('x^50); concat([0], Vec(serlaplace(- exp(2*x)*lambertw(-x) ))) \\ G. C. Greubel, Nov 08 2017

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

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

A277481 E.g.f.: -log(1+x)*LambertW(-x).

Original entry on oeis.org

0, 0, 2, 3, 32, 240, 3114, 44065, 777720, 15582168, 357427770, 9151281293, 259607392164, 8070381333872, 272960010908662, 9976300661919345, 391837137436921072, 16458193396472517328, 736145006794621566642, 34932117830021859779517, 1752782822664497750549660

Offset: 0

Vaclav Kotesovec, Oct 17 2016

nonn

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

Cf. A000169, A277465, A277473, A277482.

CoefficientList[Series[-Log[1+x]*LambertW[-x], {x, 0, 20}], x] * Range[0, 20]!
Table[n!*Sum[(-1)^(n-k-1)*k^(k-1)/(k!*(n-k)), {k, 1, n-1}], {n, 0, 20}] (* Vaclav Kotesovec, Oct 28 2016 *)

x='x+O('x^50); concat([0,0], Vec(serlaplace(-log(1+x)*lambertw(-x) ))) \\ G. C. Greubel, Nov 08 2017

a(n) ~ log(1+exp(-1)) * n^(n-1).

A277482 E.g.f.: log(1-x)*LambertW(-x).

Original entry on oeis.org

0, 0, 2, 9, 56, 480, 5394, 75775, 1280376, 25270056, 569899770, 14444562803, 406204015524, 12545427045008, 422007399953398, 15354968442741135, 600807449737710832, 25153741340051795248, 1121917008608064151218, 53107023489332468636739, 2658946993059795072656540

Offset: 0

Vaclav Kotesovec, Oct 17 2016

nonn

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

Cf. A000169, A277466, A277473, A277481.

CoefficientList[Series[Log[1-x]*LambertW[-x], {x, 0, 20}], x] * Range[0, 20]!
Table[n!*Sum[k^(k-1)/(k!*(n-k)), {k, 1, n-1}], {n, 0, 20}] (* Vaclav Kotesovec, Oct 28 2016 *)

x='x+O('x^50); concat([0,0], Vec(serlaplace(log(1-x)*lambertw(-x)) )) \\ G. C. Greubel, Nov 09 2017

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

A277484 E.g.f.: -arcsinh(x)*LambertW(-x).

Original entry on oeis.org

0, 0, 2, 6, 32, 300, 3624, 52570, 908928, 18277560, 417634080, 10682763278, 302517156864, 9394763009060, 317429118686848, 11592017133950370, 454961391572119552, 19097430979664893168, 853711115246721262080, 40490675416206345889686, 2030782746261324446228480

Offset: 0

Vaclav Kotesovec, Oct 17 2016

nonn

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

Cf. A000169, A277470, A277473, A277483.

CoefficientList[Series[-ArcSinh[x]*LambertW[-x], {x, 0, 20}], x] * Range[0, 20]!
Flatten[{0, Table[Sum[Sin[Pi*k/2] * Binomial[n, k] * (k-2)!!^2 * (n-k)^(n-k-1), {k, 1, n-1}], {n, 1, 20}]}] (* Vaclav Kotesovec, Oct 28 2016 *)

x='x+O('x^10); concat([0,0], Vec(serlaplace(-asinh(x)*lambertw(-x) ))) \\ G. C. Greubel, Nov 08 2017

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

a(n) ~ (-1 + log(1 + sqrt(1+exp(2)))) * n^(n-1).

A294411 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of e.g.f. -exp(kx)LambertW(-x).

Original entry on oeis.org

0, 0, 1, 0, 1, 2, 0, 1, 4, 9, 0, 1, 6, 18, 64, 0, 1, 8, 33, 116, 625, 0, 1, 10, 54, 216, 1060, 7776, 0, 1, 12, 81, 388, 1865, 12702, 117649, 0, 1, 14, 114, 656, 3340, 21228, 187810, 2097152, 0, 1, 16, 153, 1044, 5905, 36414, 303765, 3296120, 43046721, 0, 1, 18, 198, 1576, 10100, 63480, 500374, 5222864, 66897288, 1000000000

Offset: 0

Ilya Gutkovskiy, Oct 30 2017

			E.g.f. of column k: A_k(x) = x/1! + 2*(k + 1)*x^2/2! + 3*(k^2 + 2*k + 3)*x^3/3! + 4*(k^3 + 3*k^2 + 9*k + 16)*x^4/4! + ...
Square array begins:
    0,     0,     0,     0,     0,      0, ...
    1,     1,     1,     1,     1,      1, ...
    2,     4,     6,     8,    10,     12, ...
    9,    18,    33,    54,    81,    114, ...
   64,   116,   216,   388,   656,   1044, ...
  625,  1060,  1895,  3340,  5905,  10100, ...

G. C. Greubel, Table of n, a(n) for the first 50 antidiagonals, flattened

Columns k=0..2 give A000169, A277473, A277485.
Main diagonal gives A292633.
Cf. A290824.

Table[Function[k, n! SeriesCoefficient[-Exp[k x] LambertW[-x], {x, 0, n}]][j - n], {j, 0, 10}, {n, 0, j}] // Flatten

E.g.f. of column k: -exp(k*x)*LambertW(-x).

A372333 Expansion of e.g.f. -exp(x) * LambertW(-2*x)/2.

Original entry on oeis.org

0, 1, 6, 51, 684, 12965, 317298, 9500631, 336237016, 13729172553, 635237632350, 32844916975739, 1876755685038468, 117437155609780461, 7986793018367861194, 586578825469711599135, 46268265552518066488752, 3901008402618593931019409

Offset: 0

Seiichi Manyama, Apr 28 2024

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..352
Eric Weisstein's World of Mathematics, Lambert W-Function.

Cf. A277473, A372334.

Cf. A360548, A372315.

my(N=20, x='x+O('x^N)); concat(0, Vec(serlaplace(-exp(x)*lambertw(-2*x)/2)))

a(n) = sum(k=1, n, (2*k)^(k-1)*binomial(n, k));

a(n) = Sum_{k=1..n} (2*k)^(k-1) * binomial(n,k).
G.f.: Sum_{k>=1} (2*k)^(k-1) * x^k / (1-x)^(k+1).
a(n) ~ exp(exp(-1)/2) * 2^(n-1) * n^(n-1). - Vaclav Kotesovec, Apr 30 2024

A372334 Expansion of e.g.f. -exp(x) * LambertW(-3*x)/3.

Original entry on oeis.org

0, 1, 8, 102, 2092, 60140, 2220954, 100119670, 5328468968, 326960686872, 22724388453070, 1764411577328906, 151364204180518476, 14217940294767407380, 1451334877597451677250, 159972528561402504191190, 18936257811933773637390544, 2395818853376147403857700656

Offset: 0

Seiichi Manyama, Apr 28 2024

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..334
Eric Weisstein's World of Mathematics, Lambert W-Function.

Cf. A277473, A372333.

Cf. A372316.

my(N=20, x='x+O('x^N)); concat(0, Vec(serlaplace(-exp(x)*lambertw(-3*x)/3)))

a(n) = sum(k=1, n, (3*k)^(k-1)*binomial(n, k));

a(n) = Sum_{k=1..n} (3*k)^(k-1) * binomial(n,k).
G.f.: Sum_{k>=1} (3*k)^(k-1) * x^k / (1-x)^(k+1).
a(n) ~ exp(exp(-1)/3) * 3^(n-1) * n^(n-1). - Vaclav Kotesovec, Apr 30 2024

cp's OEIS Frontend

A277483 E.g.f.: -arcsin(x)*LambertW(-x).

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A277485 E.g.f.: -exp(2*x)*LambertW(-x).

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A277481 E.g.f.: -log(1+x)*LambertW(-x).

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A277482 E.g.f.: log(1-x)*LambertW(-x).

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A277484 E.g.f.: -arcsinh(x)*LambertW(-x).

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A294411 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of e.g.f. -exp(k*x)*LambertW(-x).

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A372333 Expansion of e.g.f. -exp(x) * LambertW(-2*x)/2.

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PARI

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A372334 Expansion of e.g.f. -exp(x) * LambertW(-3*x)/3.

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PARI

A277485 E.g.f.: -exp(2x)LambertW(-x).

A294411 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of e.g.f. -exp(kx)LambertW(-x).